💡First, I recognize that the problem involves finding the area of a square when its perimeter is given.I know that the perimeter of a square is calculated by adding up the lengths of all four sides, which are equal in a square. Therefore, if the perimeter is 32 feet, each side must be 32 divided by 4, which is 8 feet.Once I have the length of one side, I can find the area of the square by squaring the length of one side. Squaring 8 feet gives me 64 square feet.So, the area of the square is 64 square feet.

💡First, I need to determine the range of possible values for ( x ) in the triangle with sides ( x ), 5, and 6. I'll use the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the remaining side.1. Applying the theorem to sides ( x ) and 5: [ x + 5 > 6 implies x > 1 ]2. Applying the theorem to sides ( x ) and 6: [ x + 6 > 5 implies x > -1 ] Since ( x ) must be positive, this doesn't provide a new constraint.3. Applying the theorem to sides 5 and 6: [ 5 + 6 > x implies 11 > x ]Combining the inequalities from steps 1 and 3, we find:[1 < x < 11]Since ( x ) must be an integer, the possible values of ( x ) are 2, 3, 4, 5, 6, 7, 8, 9, and 10.The greatest possible integral value of ( x ) is 10, and the least possible integral value is 2. The positive difference between these values is:[10 - 2 = 8]

💡Okay, so I need to find the degree of this polynomial that's the product of three other polynomials: (x⁵ + a x⁸ + b x² + c), (x⁴ + d x³ + e x + f), and (x² + g). All the constants a through g are nonzero. Hmm, I remember that the degree of a polynomial is the highest power of x with a nonzero coefficient. So, when you multiply polynomials together, the degree of the resulting polynomial is the sum of the degrees of the individual polynomials, right?Let me double-check that. If I have two polynomials, say P(x) of degree m and Q(x) of degree n, then when I multiply them, the highest degree term will be x^m * x^n = x^(m+n). So yes, the degree of the product is m + n. So, applying that to three polynomials, it should be the sum of the degrees of all three.First, I need to find the degree of each polynomial individually. Let's start with the first one: (x⁵ + a x⁸ + b x² + c). The terms here are x⁵, a x⁸, b x², and c. The degrees of these terms are 5, 8, 2, and 0 (since c is a constant). The highest degree here is 8, so the degree of the first polynomial is 8.Next, the second polynomial: (x⁴ + d x³ + e x + f). The terms are x⁴, d x³, e x, and f. Their degrees are 4, 3, 1, and 0. The highest degree here is 4, so the degree of the second polynomial is 4.Lastly, the third polynomial: (x² + g). The terms are x² and g. Their degrees are 2 and 0. The highest degree is 2, so the degree of the third polynomial is 2.Now, adding these degrees together: 8 (from the first polynomial) + 4 (from the second) + 2 (from the third) = 14. So, the degree of the product polynomial should be 14.Wait, let me make sure I'm not missing anything. Sometimes, when multiplying polynomials, the leading terms might cancel out if their coefficients are such, but in this case, all the constants a through g are nonzero. So, the leading term will definitely be a x⁸ * x⁴ * x² = a x¹⁴, which is nonzero because a is nonzero. Therefore, the degree is indeed 14.I think that's it. The degree of the resulting polynomial is 14.

💡Okay, so I have this problem about a geometric sequence. The first three terms are given as ( a - 2 ), ( 4 ), and ( 2a ). All terms are positive, which is important. I need to find the sum of the first ( n ) terms of another sequence ( {b_n} ), where each ( b_n = (2n - 1)S_n ), and ( S_n ) is the sum of the first ( n ) terms of the geometric sequence. The goal is to find ( T_n ), the sum of the first ( n ) terms of ( {b_n} ).First, let me recall what a geometric sequence is. It's a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio, usually denoted as ( r ). So, if the first term is ( a_1 ), then the terms are ( a_1, a_1 r, a_1 r^2, ldots ).Given that the first three terms are ( a - 2 ), ( 4 ), and ( 2a ), I can set up the relationships based on the common ratio. In a geometric sequence, the ratio between consecutive terms is constant. So, the ratio between the second and first term should be equal to the ratio between the third and second term.Mathematically, that means:[frac{4}{a - 2} = frac{2a}{4}]Simplifying this equation should give me the value of ( a ).Let me compute that step by step. Cross-multiplying gives:[4 times 4 = (a - 2) times 2a]Which simplifies to:[16 = 2a(a - 2)]Dividing both sides by 2:[8 = a(a - 2)]Expanding the right side:[8 = a^2 - 2a]Bringing all terms to one side:[a^2 - 2a - 8 = 0]Now, solving this quadratic equation. The quadratic formula is ( a = frac{-b pm sqrt{b^2 - 4ac}}{2a} ), where in this case, ( a = 1 ), ( b = -2 ), and ( c = -8 ).Plugging in the values:[a = frac{-(-2) pm sqrt{(-2)^2 - 4 times 1 times (-8)}}{2 times 1} = frac{2 pm sqrt{4 + 32}}{2} = frac{2 pm sqrt{36}}{2} = frac{2 pm 6}{2}]So, the solutions are:[a = frac{2 + 6}{2} = 4 quad text{and} quad a = frac{2 - 6}{2} = -2]But since all terms of the geometric sequence are positive, ( a - 2 ) must be positive. If ( a = -2 ), then ( a - 2 = -4 ), which is negative. Therefore, ( a = 4 ) is the valid solution.So, the first term is ( a - 2 = 4 - 2 = 2 ), the second term is ( 4 ), and the third term is ( 2a = 8 ). Therefore, the geometric sequence is ( 2, 4, 8, ldots ), which is a geometric sequence with the first term ( a_1 = 2 ) and common ratio ( r = 2 ).Next, I need to find ( S_n ), the sum of the first ( n ) terms of this geometric sequence. The formula for the sum of the first ( n ) terms of a geometric sequence is:[S_n = a_1 frac{r^n - 1}{r - 1}]Plugging in the values ( a_1 = 2 ) and ( r = 2 ):[S_n = 2 frac{2^n - 1}{2 - 1} = 2(2^n - 1) = 2^{n+1} - 2]So, ( S_n = 2^{n+1} - 2 ).Now, the sequence ( {b_n} ) is defined as ( b_n = (2n - 1)S_n ). Therefore, each term ( b_n ) is ( (2n - 1)(2^{n+1} - 2) ).The problem asks for ( T_n ), which is the sum of the first ( n ) terms of ( {b_n} ). So, ( T_n = sum_{k=1}^{n} b_k = sum_{k=1}^{n} (2k - 1)(2^{k+1} - 2) ).Let me write this out explicitly:[T_n = sum_{k=1}^{n} (2k - 1)(2^{k+1} - 2)]I can expand this product inside the summation:[(2k - 1)(2^{k+1} - 2) = (2k - 1)2^{k+1} - (2k - 1)2]So, the sum becomes:[T_n = sum_{k=1}^{n} left[ (2k - 1)2^{k+1} - (2k - 1)2 right] = sum_{k=1}^{n} (2k - 1)2^{k+1} - sum_{k=1}^{n} (2k - 1)2]Let me denote the first sum as ( C_n = sum_{k=1}^{n} (2k - 1)2^{k+1} ) and the second sum as ( D_n = sum_{k=1}^{n} (2k - 1)2 ). Then, ( T_n = C_n - D_n ).First, let me compute ( D_n ) because it seems simpler. ( D_n = sum_{k=1}^{n} (2k - 1)2 ). Factor out the 2:[D_n = 2 sum_{k=1}^{n} (2k - 1)]The sum ( sum_{k=1}^{n} (2k - 1) ) is a known arithmetic series. The terms are ( 1, 3, 5, ldots, (2n - 1) ), which is the sum of the first ( n ) odd numbers. The sum of the first ( n ) odd numbers is ( n^2 ). Therefore:[D_n = 2 times n^2 = 2n^2]So, ( D_n = 2n^2 ).Now, let's tackle ( C_n = sum_{k=1}^{n} (2k - 1)2^{k+1} ). This seems more complicated. Let me write it as:[C_n = sum_{k=1}^{n} (2k - 1)2^{k+1} = 2 sum_{k=1}^{n} (2k - 1)2^{k}]Wait, that might not help directly. Alternatively, perhaps I can find a recursive formula or use generating functions. Alternatively, I can look for a pattern or use the method of differences.Let me consider the general term ( (2k - 1)2^{k+1} ). Let me denote ( T_k = (2k - 1)2^{k+1} ). Then, ( C_n = sum_{k=1}^{n} T_k ).To compute ( C_n ), perhaps I can find a recurrence relation or find a closed-form expression by manipulating the series.Let me consider the sum ( C_n = sum_{k=1}^{n} (2k - 1)2^{k+1} ). Let me factor out the 2^{k+1}:[C_n = sum_{k=1}^{n} (2k - 1)2^{k+1} = 2 sum_{k=1}^{n} (2k - 1)2^{k}]Wait, that's the same as before. Maybe I can write ( (2k - 1)2^{k} ) as ( 2^{k+1}k - 2^{k} ). Let me see:[(2k - 1)2^{k} = 2^{k+1}k - 2^{k}]Yes, that's correct. So, substituting back:[C_n = 2 sum_{k=1}^{n} (2^{k+1}k - 2^{k}) = 2 left( sum_{k=1}^{n} 2^{k+1}k - sum_{k=1}^{n} 2^{k} right )]Simplify each sum separately.First, compute ( sum_{k=1}^{n} 2^{k+1}k ). Let me denote this as ( S_1 = sum_{k=1}^{n} 2^{k+1}k ). Similarly, ( sum_{k=1}^{n} 2^{k} ) is a geometric series, which I can compute.Let me compute ( S_1 ). Notice that ( 2^{k+1}k = 2 times 2^{k}k ). So, ( S_1 = 2 sum_{k=1}^{n} 2^{k}k ).The sum ( sum_{k=1}^{n} k r^k ) is a known series, and its closed-form is ( r frac{1 - (n+1)r^n + n r^{n+1}}{(1 - r)^2} ) for ( r neq 1 ). In this case, ( r = 2 ), so plugging in:[sum_{k=1}^{n} k 2^k = 2 frac{1 - (n+1)2^n + n 2^{n+1}}{(1 - 2)^2} = 2 frac{1 - (n+1)2^n + n 2^{n+1}}{1}]Simplify the numerator:[1 - (n+1)2^n + n 2^{n+1} = 1 - (n+1)2^n + 2n 2^n = 1 + (2n - n - 1)2^n = 1 + (n - 1)2^n]Therefore:[sum_{k=1}^{n} k 2^k = 2(1 + (n - 1)2^n) = 2 + (n - 1)2^{n+1}]So, ( S_1 = 2 times [2 + (n - 1)2^{n+1}] = 4 + (n - 1)2^{n+2} ).Now, compute ( sum_{k=1}^{n} 2^{k} ). This is a geometric series with first term ( 2 ) and ratio ( 2 ). The sum is:[sum_{k=1}^{n} 2^{k} = 2(2^n - 1)/(2 - 1) = 2(2^n - 1) = 2^{n+1} - 2]So, putting it all together:[C_n = 2 left( S_1 - sum_{k=1}^{n} 2^{k} right ) = 2 left( [4 + (n - 1)2^{n+2}] - [2^{n+1} - 2] right )]Simplify inside the brackets:[4 + (n - 1)2^{n+2} - 2^{n+1} + 2 = (4 + 2) + (n - 1)2^{n+2} - 2^{n+1} = 6 + (n - 1)2^{n+2} - 2^{n+1}]Factor ( 2^{n+1} ) from the last two terms:[6 + 2^{n+1} [ (n - 1)2 - 1 ] = 6 + 2^{n+1}(2n - 2 - 1) = 6 + 2^{n+1}(2n - 3)]Therefore:[C_n = 2 times [6 + 2^{n+1}(2n - 3)] = 12 + 2^{n+2}(2n - 3)]So, ( C_n = (2n - 3)2^{n+2} + 12 ).Now, recall that ( T_n = C_n - D_n ), and ( D_n = 2n^2 ). Therefore:[T_n = (2n - 3)2^{n+2} + 12 - 2n^2]So, the sum of the first ( n ) terms of the sequence ( {b_n} ) is ( T_n = (2n - 3)2^{n+2} + 12 - 2n^2 ).Let me double-check my steps to ensure I didn't make any mistakes.1. Found ( a = 4 ) correctly by solving the quadratic equation and considering positivity.2. Calculated ( S_n = 2^{n+1} - 2 ) correctly.3. Expressed ( b_n = (2n - 1)S_n ) correctly.4. Expanded ( T_n = sum (2k -1)(2^{k+1} - 2) ) correctly into two sums ( C_n ) and ( D_n ).5. Computed ( D_n = 2n^2 ) correctly by recognizing the sum of first ( n ) odd numbers.6. Broke down ( C_n ) into two sums, used the known formula for ( sum k 2^k ), computed ( S_1 ) correctly, and then substituted back.7. Simplified the expressions step by step, ensuring each algebraic manipulation was correct.I think all steps are correct. So, the final expression for ( T_n ) is ( (2n - 3)2^{n+2} + 12 - 2n^2 ).**Final Answer**The sum of the first ( n ) terms of the sequence ({b_n}) is boxed{(2n - 3) cdot 2^{n+2} + 12 - 2n^2}.

💡Okay, so I have this inequality to solve: (frac{x(x+2)}{(x-3)^2} ge 8). Hmm, okay, let me try to figure this out step by step. First, I remember that when dealing with inequalities involving fractions, I need to be careful about the denominator because it can't be zero, and its sign affects the inequality. So, the denominator here is ((x-3)^2). Since it's squared, it's always non-negative, right? That means it's either positive or zero. But wait, if (x = 3), the denominator becomes zero, which is undefined. So, (x) can't be 3. I'll keep that in mind.Now, since the denominator is always positive except when (x = 3), I can multiply both sides of the inequality by ((x-3)^2) without changing the direction of the inequality. That's good because it simplifies things. So, multiplying both sides by ((x-3)^2), I get:(x(x + 2) ge 8(x - 3)^2)Alright, now I need to expand both sides to simplify this inequality. Let's start with the left side:(x(x + 2) = x^2 + 2x)Okay, that was straightforward. Now, the right side is (8(x - 3)^2). Let me expand that:First, expand ((x - 3)^2):((x - 3)^2 = x^2 - 6x + 9)So, multiplying by 8:(8(x^2 - 6x + 9) = 8x^2 - 48x + 72)Now, putting it all back into the inequality:(x^2 + 2x ge 8x^2 - 48x + 72)Hmm, I need to bring all terms to one side to solve this inequality. Let me subtract (8x^2 - 48x + 72) from both sides:(x^2 + 2x - 8x^2 + 48x - 72 ge 0)Combine like terms:(x^2 - 8x^2 = -7x^2)(2x + 48x = 50x)So, putting it all together:(-7x^2 + 50x - 72 ge 0)Hmm, I have a quadratic inequality here. It's a bit messy with the negative coefficient on (x^2). Maybe I can multiply both sides by -1 to make it positive. But wait, multiplying both sides of an inequality by a negative number flips the inequality sign. So, let's do that:Multiplying by -1:(7x^2 - 50x + 72 le 0)Okay, now it's (7x^2 - 50x + 72 le 0). That looks better. Now, I need to find the values of (x) where this quadratic expression is less than or equal to zero. To do that, I should find the roots of the quadratic equation (7x^2 - 50x + 72 = 0).Let me use the quadratic formula. The quadratic formula is (x = frac{-b pm sqrt{b^2 - 4ac}}{2a}), where (a = 7), (b = -50), and (c = 72).Plugging in the values:(x = frac{-(-50) pm sqrt{(-50)^2 - 4 times 7 times 72}}{2 times 7})Simplify:(x = frac{50 pm sqrt{2500 - 2016}}{14})Calculate the discriminant:(2500 - 2016 = 484)So, (sqrt{484} = 22)Therefore, the roots are:(x = frac{50 pm 22}{14})Calculating both possibilities:First root:(x = frac{50 + 22}{14} = frac{72}{14} = frac{36}{7} approx 5.14)Second root:(x = frac{50 - 22}{14} = frac{28}{14} = 2)So, the roots are (x = 2) and (x = frac{36}{7}) (which is approximately 5.14). Wait, hold on, that doesn't seem right because earlier when I expanded, I think I might have made a mistake in the coefficients.Wait, let me double-check my earlier steps. When I expanded (x(x + 2)), I got (x^2 + 2x), which is correct. Then, expanding (8(x - 3)^2) gave me (8x^2 - 48x + 72), which also seems correct.Subtracting (8x^2 - 48x + 72) from both sides:(x^2 + 2x - 8x^2 + 48x - 72 ge 0)Which simplifies to:(-7x^2 + 50x - 72 ge 0)Multiplying by -1:(7x^2 - 50x + 72 le 0)So, the quadratic is (7x^2 - 50x + 72). Let me recalculate the discriminant to make sure I didn't make a mistake there.Discriminant (D = b^2 - 4ac = (-50)^2 - 4 times 7 times 72 = 2500 - 2016 = 484). That's correct, and (sqrt{484} = 22).So, roots are:(x = frac{50 pm 22}{14})Calculating again:First root: (frac{50 + 22}{14} = frac{72}{14} = frac{36}{7} approx 5.14)Second root: (frac{50 - 22}{14} = frac{28}{14} = 2)Wait, so the roots are at (x = 2) and (x = frac{36}{7}). Hmm, that seems a bit off because when I initially thought, I thought the roots were 2 and 48, but that must have been a mistake in my earlier calculation.Wait, let me check the expansion again. Maybe I messed up when expanding (8(x - 3)^2). Let me redo that.((x - 3)^2 = x^2 - 6x + 9). Multiplying by 8: (8x^2 - 48x + 72). That's correct.So, the left side is (x^2 + 2x), and the right side is (8x^2 - 48x + 72). Subtracting the right side from the left:(x^2 + 2x - 8x^2 + 48x - 72 = -7x^2 + 50x - 72). That's correct.So, the quadratic is indeed (7x^2 - 50x + 72 le 0), with roots at (x = 2) and (x = frac{36}{7}).Wait, but (frac{36}{7}) is approximately 5.14, which is much smaller than 48. So, where did I get 48 from earlier? Maybe I made a mistake in my initial thought process.Wait, in the initial problem, the user's solution had roots at 2 and 48, but that seems incorrect. Let me see.Wait, no, actually, in the initial problem, the user's solution had roots at 2 and 48, but that was probably a mistake because when I did the calculation, the roots are at 2 and (frac{36}{7}).Wait, maybe I should double-check the quadratic equation.Wait, let me write the quadratic inequality again:(7x^2 - 50x + 72 le 0)So, the quadratic is (7x^2 - 50x + 72). Let me factor this quadratic to see if it can be factored.Looking for two numbers that multiply to (7 times 72 = 504) and add up to -50.Hmm, factors of 504: 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 18, 21, 24, 28, 36, 42, 56, 63, 72, 84, 126, 168, 252, 504.Looking for two numbers that add up to -50. Let's see:- Let's try -18 and -32: 18 + 32 = 50, but 18 * 32 = 576, which is too big.Wait, maybe -14 and -36: 14 + 36 = 50, and 14 * 36 = 504. Yes, that works.So, we can write the quadratic as:(7x^2 - 14x - 36x + 72)Grouping terms:((7x^2 - 14x) + (-36x + 72))Factor each group:(7x(x - 2) - 36(x - 2))Now, factor out the common term ((x - 2)):((7x - 36)(x - 2))So, the quadratic factors as ((7x - 36)(x - 2)). Therefore, the roots are (x = 2) and (x = frac{36}{7}), which is approximately 5.14.Okay, so the quadratic (7x^2 - 50x + 72) factors to ((7x - 36)(x - 2)), and the roots are at (x = 2) and (x = frac{36}{7}).Now, since the quadratic opens upwards (because the coefficient of (x^2) is positive), the quadratic will be less than or equal to zero between its roots. So, the solution to (7x^2 - 50x + 72 le 0) is (2 le x le frac{36}{7}).But wait, we have to remember that (x neq 3) because the original inequality has a denominator of ((x - 3)^2), which is zero when (x = 3). So, we need to exclude (x = 3) from our solution.So, the interval (2 le x le frac{36}{7}) includes (x = 3), which we need to exclude. Therefore, the solution is (2 le x < 3) or (3 < x le frac{36}{7}).But wait, (frac{36}{7}) is approximately 5.14, which is less than 48. So, the initial solution that the user had with 48 was incorrect.Wait, maybe I made a mistake in the initial expansion. Let me double-check.Original inequality: (frac{x(x + 2)}{(x - 3)^2} ge 8)Multiplying both sides by ((x - 3)^2), which is positive except at (x = 3), so inequality remains the same:(x(x + 2) ge 8(x - 3)^2)Expanding left side: (x^2 + 2x)Expanding right side: (8(x^2 - 6x + 9) = 8x^2 - 48x + 72)Bringing all terms to left:(x^2 + 2x - 8x^2 + 48x - 72 ge 0)Simplify:(-7x^2 + 50x - 72 ge 0)Multiplying by -1:(7x^2 - 50x + 72 le 0)Factoring: ((7x - 36)(x - 2) le 0)So, roots at (x = 2) and (x = frac{36}{7}), approximately 5.14.Therefore, the solution is (2 le x le frac{36}{7}), excluding (x = 3).So, in interval notation, that would be ([2, 3) cup (3, frac{36}{7}]).Wait, but (frac{36}{7}) is approximately 5.14, so the upper bound is around 5.14, not 48. So, the initial solution that the user had with 48 was incorrect.Therefore, the correct solution is (x) in the interval ([2, 3) cup (3, frac{36}{7}]).But wait, let me double-check the quadratic equation again to make sure I didn't make a mistake in factoring.Quadratic: (7x^2 - 50x + 72)Factored as ((7x - 36)(x - 2)). Let me multiply it back to confirm:((7x - 36)(x - 2) = 7x(x) + 7x(-2) - 36(x) + (-36)(-2) = 7x^2 - 14x - 36x + 72 = 7x^2 - 50x + 72). Yes, that's correct.So, the roots are indeed at (x = 2) and (x = frac{36}{7}).Therefore, the solution to the inequality (7x^2 - 50x + 72 le 0) is (2 le x le frac{36}{7}).But since (x = 3) is excluded, the solution is (2 le x < 3) or (3 < x le frac{36}{7}).So, in interval notation, that's ([2, 3) cup (3, frac{36}{7}]).Wait, but (frac{36}{7}) is approximately 5.14, so the upper bound is around 5.14, not 48. So, the initial solution that the user had with 48 was incorrect.Therefore, the correct solution is (x) in the interval ([2, 3) cup (3, frac{36}{7}]).But wait, let me check if I made a mistake in the initial expansion. Maybe I missed a term.Original inequality: (frac{x(x + 2)}{(x - 3)^2} ge 8)Multiplying both sides by ((x - 3)^2):(x(x + 2) ge 8(x - 3)^2)Expanding left side: (x^2 + 2x)Expanding right side: (8(x^2 - 6x + 9) = 8x^2 - 48x + 72)Bringing all terms to left:(x^2 + 2x - 8x^2 + 48x - 72 ge 0)Simplify:(-7x^2 + 50x - 72 ge 0)Multiplying by -1:(7x^2 - 50x + 72 le 0)Factoring: ((7x - 36)(x - 2) le 0)Roots at (x = 2) and (x = frac{36}{7}), approximately 5.14.So, the solution is (2 le x le frac{36}{7}), excluding (x = 3).Therefore, the correct interval is ([2, 3) cup (3, frac{36}{7}]).Wait, but in the initial problem, the user had the solution as ([2, 3) cup (3, 48]), which is incorrect because the upper bound should be (frac{36}{7}), not 48.So, I think the initial solution had a mistake in the quadratic expansion or factoring.Therefore, the correct solution is ([2, 3) cup (3, frac{36}{7}]).But let me confirm by testing a value in each interval to make sure.The critical points are (x = 2), (x = 3), and (x = frac{36}{7}) (approximately 5.14).So, the intervals to test are:1. (x < 2)2. (2 < x < 3)3. (3 < x < frac{36}{7})4. (x > frac{36}{7})Let's pick test points in each interval.1. (x = 0): Plug into the original inequality: (frac{0(0 + 2)}{(0 - 3)^2} = 0). Is (0 ge 8)? No. So, this interval is not part of the solution.2. (x = 2.5): Plug into the original inequality.Calculate numerator: (2.5(2.5 + 2) = 2.5 times 4.5 = 11.25)Denominator: ((2.5 - 3)^2 = (-0.5)^2 = 0.25)So, (frac{11.25}{0.25} = 45). Is (45 ge 8)? Yes. So, this interval is part of the solution.3. (x = 4): Plug into the original inequality.Numerator: (4(4 + 2) = 4 times 6 = 24)Denominator: ((4 - 3)^2 = 1^2 = 1)So, (frac{24}{1} = 24). Is (24 ge 8)? Yes. So, this interval is part of the solution.4. (x = 6): Plug into the original inequality.Numerator: (6(6 + 2) = 6 times 8 = 48)Denominator: ((6 - 3)^2 = 3^2 = 9)So, (frac{48}{9} approx 5.33). Is (5.33 ge 8)? No. So, this interval is not part of the solution.Therefore, the solution is (2 le x < 3) and (3 < x le frac{36}{7}), which is approximately 5.14.So, in interval notation, that's ([2, 3) cup (3, frac{36}{7}]).Wait, but (frac{36}{7}) is approximately 5.14, so the upper bound is around 5.14, not 48. So, the initial solution that the user had with 48 was incorrect.Therefore, the correct solution is ([2, 3) cup (3, frac{36}{7}]).But wait, let me make sure I didn't make a mistake in the quadratic equation.Quadratic equation: (7x^2 - 50x + 72 = 0)Using quadratic formula:(x = frac{50 pm sqrt{2500 - 2016}}{14} = frac{50 pm sqrt{484}}{14} = frac{50 pm 22}{14})So, (x = frac{72}{14} = frac{36}{7}) and (x = frac{28}{14} = 2). Correct.So, the roots are indeed at (x = 2) and (x = frac{36}{7}).Therefore, the solution is ([2, 3) cup (3, frac{36}{7}]).I think that's correct. So, the initial solution with 48 was wrong, and the correct upper bound is (frac{36}{7}).

💡Okay, so I have this problem about a curve C, which is the locus of points where the sum of the distances to three fixed points F1(-1,0), F2(1,0), and F3(0,1) is equal to 2√2. There are four conclusions given, and I need to figure out which ones are true. Let me try to break this down step by step.First, let me visualize the three fixed points. F1 is at (-1,0), F2 is at (1,0), and F3 is at (0,1). So, F1 and F2 are on the x-axis, symmetric about the origin, and F3 is on the y-axis above the origin. The curve C consists of all points P such that the sum of the distances from P to each of these three points is 2√2.I remember that when dealing with the sum of distances to fixed points, it's related to ellipses, but usually, an ellipse is defined as the set of points where the sum of distances to two foci is constant. Here, we have three foci, so it's a different kind of curve. Maybe it's a type of polyellipse or something else. I'm not entirely sure, but I'll keep that in mind.Let me start by analyzing each conclusion one by one.**Conclusion ①: Curve C is symmetrical about both the x-axis and the y-axis.**Hmm, symmetry. If the curve is symmetrical about the x-axis, then for any point (x, y) on the curve, the point (x, -y) should also be on the curve. Similarly, if it's symmetrical about the y-axis, then for any point (x, y), the point (-x, y) should also be on the curve.Let me test this. Suppose P(x, y) is on curve C. Then, the sum of distances from P to F1, F2, and F3 is 2√2. If I reflect P over the x-axis to get P'(x, -y), what happens to the distances?- Distance from P' to F1: √[(x + 1)^2 + (-y)^2] = √[(x + 1)^2 + y^2], which is the same as the distance from P to F1.- Similarly, distance from P' to F2: √[(x - 1)^2 + (-y)^2] = √[(x - 1)^2 + y^2], same as P to F2.- Distance from P' to F3: √[x^2 + (-y - 1)^2] = √[x^2 + (y + 1)^2], which is different from the distance from P to F3, which is √[x^2 + (y - 1)^2].So, unless y = 0, the distance to F3 changes. Therefore, unless P is on the x-axis, reflecting over the x-axis changes the distance to F3. Hence, the curve is not symmetrical about the x-axis.What about the y-axis? Let's reflect P(x, y) over the y-axis to get P'(-x, y).- Distance from P' to F1: √[(-x + 1)^2 + y^2] = √[(x - 1)^2 + y^2], which is the same as the distance from P to F2.- Distance from P' to F2: √[(-x - 1)^2 + y^2] = √[(x + 1)^2 + y^2], same as the distance from P to F1.- Distance from P' to F3: √[(-x)^2 + (y - 1)^2] = √[x^2 + (y - 1)^2], same as the distance from P to F3.So, the sum of distances for P' is the same as for P, just swapping the distances to F1 and F2. Therefore, if P is on curve C, then P' is also on curve C. So, the curve is symmetrical about the y-axis.But since it's not symmetrical about the x-axis, conclusion ① is incorrect.**Conclusion ②: There exists a point P on curve C such that |PF3| = 2√2 / 3.**Okay, so we need to check if there's a point P on C where the distance from P to F3 is exactly 2√2 / 3.Let me denote |PF1| + |PF2| + |PF3| = 2√2. If |PF3| = 2√2 / 3, then |PF1| + |PF2| = 2√2 - 2√2 / 3 = 4√2 / 3.Now, I know that for any point P, the sum |PF1| + |PF2| is at least the distance between F1 and F2, which is |F1F2|. Let me compute |F1F2|.F1 is (-1,0) and F2 is (1,0), so the distance between them is √[(1 - (-1))^2 + (0 - 0)^2] = √[4] = 2.So, |PF1| + |PF2| ≥ 2. But in this case, |PF1| + |PF2| = 4√2 / 3 ≈ 1.885, which is less than 2. That's impossible because the sum of distances from any point to F1 and F2 cannot be less than the distance between F1 and F2.Therefore, such a point P cannot exist. So, conclusion ② is incorrect.**Conclusion ③: If point P is on curve C, then the maximum area of triangle F1PF2 is 1.**Alright, I need to find the maximum area of triangle F1PF2 where P is on curve C.First, let's recall that the area of triangle F1PF2 can be calculated using the formula:Area = (1/2) * |F1F2| * height,where height is the distance from point P to the line F1F2.Since F1 and F2 are on the x-axis, the line F1F2 is the x-axis itself. So, the height is just the y-coordinate of P, which is |y|.Therefore, the area is (1/2) * 2 * |y| = |y|.So, to maximize the area, we need to maximize |y|, i.e., find the maximum y-coordinate of a point P on curve C.So, let's try to find the maximum y on curve C.Given that |PF1| + |PF2| + |PF3| = 2√2.We can write this as:√[(x + 1)^2 + y^2] + √[(x - 1)^2 + y^2] + √[x^2 + (y - 1)^2] = 2√2.To find the maximum y, perhaps we can consider points where x = 0 because of symmetry? Let me test that.If x = 0, then the equation becomes:√[(0 + 1)^2 + y^2] + √[(0 - 1)^2 + y^2] + √[0^2 + (y - 1)^2] = 2√2.Simplify:√(1 + y^2) + √(1 + y^2) + √[(y - 1)^2] = 2√2.So, 2√(1 + y^2) + |y - 1| = 2√2.Since we are looking for maximum y, let's assume y ≥ 1, so |y - 1| = y - 1.Then, the equation becomes:2√(1 + y^2) + y - 1 = 2√2.Let me denote this as:2√(1 + y^2) + y = 2√2 + 1.This seems a bit complicated. Maybe I can try plugging in y = 1.At y = 1:2√(1 + 1) + 1 = 2√2 + 1 ≈ 2*1.414 + 1 ≈ 2.828 + 1 = 3.828.But 2√2 + 1 ≈ 3.828, so that's equal. So, when y = 1, the equation holds.Wait, so when x = 0 and y = 1, the point P is (0,1), which is exactly F3. So, let's check:|PF1| + |PF2| + |PF3| = |F3F1| + |F3F2| + |F3F3|.Compute |F3F1|: distance from (0,1) to (-1,0): √[(0 - (-1))^2 + (1 - 0)^2] = √[1 + 1] = √2.Similarly, |F3F2|: distance from (0,1) to (1,0): √[(0 - 1)^2 + (1 - 0)^2] = √[1 + 1] = √2.And |F3F3| = 0.So, total sum is √2 + √2 + 0 = 2√2, which matches.Therefore, point (0,1) is on curve C, and its y-coordinate is 1, which gives the area of triangle F1PF2 as |y| = 1.Is this the maximum? Let's see if we can get a higher y.Suppose y > 1. Then, |y - 1| = y - 1, and the equation becomes:2√(1 + y^2) + y - 1 = 2√2.But when y = 1, the left side is 2√2 + 1 - 1 = 2√2, which is correct.If y increases beyond 1, say y = 1 + h, h > 0, then:2√(1 + (1 + h)^2) + (1 + h) - 1 = 2√2.Simplify:2√(1 + 1 + 2h + h^2) + h = 2√2.So, 2√(2 + 2h + h^2) + h = 2√2.But as h increases, √(2 + 2h + h^2) increases, and so does the left side. Therefore, the left side becomes larger than 2√2, which contradicts the equation. Hence, y cannot be greater than 1.If y < 1, then |y - 1| = 1 - y, and the equation becomes:2√(1 + y^2) + 1 - y = 2√2.Let me see if there's a solution for y < 1.Let me try y = 0:2√1 + 1 - 0 = 2*1 + 1 = 3, which is greater than 2√2 ≈ 2.828.So, at y = 0, left side is 3, which is bigger than 2√2.At y = 1, left side is 2√2 + 1 - 1 = 2√2.So, as y decreases from 1 to 0, the left side decreases from 2√2 to 3? Wait, that doesn't make sense because when y decreases, √(1 + y^2) decreases, so 2√(1 + y^2) decreases, but 1 - y increases.Wait, maybe it's not straightforward. Let me compute the derivative or see the behavior.Alternatively, maybe the maximum y is indeed 1, achieved at point (0,1). Therefore, the maximum area is 1.Hence, conclusion ③ is correct.**Conclusion ④: The maximum area of triangle PF2F3 is √3 / 2.**Alright, now I need to find the maximum area of triangle PF2F3 where P is on curve C.First, let's recall the coordinates of F2 and F3: F2 is (1,0), F3 is (0,1). So, the base of the triangle can be considered as the distance between F2 and F3.Compute |F2F3|: distance between (1,0) and (0,1) is √[(1 - 0)^2 + (0 - 1)^2] = √[1 + 1] = √2.The area of triangle PF2F3 can be calculated using the formula:Area = (1/2) * |F2F3| * height,where height is the distance from point P to the line F2F3.First, let me find the equation of the line F2F3.Points F2(1,0) and F3(0,1). The slope is (1 - 0)/(0 - 1) = -1. So, the equation is y = -x + 1.The distance from point P(x,y) to this line is | -x - y + 1 | / √(1 + 1) = | -x - y + 1 | / √2.Therefore, the area is (1/2) * √2 * | -x - y + 1 | / √2 = (1/2) * | -x - y + 1 |.So, Area = (1/2) | -x - y + 1 |.We need to maximize this area, which is equivalent to maximizing | -x - y + 1 |.Given that P(x,y) is on curve C, so √[(x + 1)^2 + y^2] + √[(x - 1)^2 + y^2] + √[x^2 + (y - 1)^2] = 2√2.This seems complicated. Maybe I can use some geometric insights.Alternatively, perhaps the maximum area occurs when P is such that the height is maximized. So, we need to maximize | -x - y + 1 |.But since we have the constraint on the sum of distances, it's not straightforward.Alternatively, maybe we can parametrize the curve or use Lagrange multipliers, but that might be too involved.Wait, let me think about the possible positions of P.If I consider the ellipse with foci at F2 and F3, with major axis length 2√2, but wait, the sum of distances to F2 and F3 would be 2√2 - |PF1|.But since |PF1| is positive, the sum |PF2| + |PF3| ≤ 2√2.But the maximum area of triangle PF2F3 would be when the height is maximum, which is when P is farthest from the line F2F3.But how far can P be from F2F3?Alternatively, maybe the maximum area is achieved when P is at a certain point, perhaps where the derivative is zero.Alternatively, maybe the maximum area is indeed √3 / 2, as given, but I need to verify.Wait, let me compute the area when P is at (0,1). Then, the area would be (1/2) | -0 -1 +1 | = (1/2)*0 = 0. So, that's not helpful.What about when P is at (1,0)? Then, the area is (1/2) | -1 -0 +1 | = 0. Similarly, at (0,0): (1/2)| -0 -0 +1 | = 1/2.Wait, so at (0,0), the area is 1/2. But is that the maximum?Wait, let me check another point. Suppose P is somewhere else.Alternatively, maybe the maximum area occurs when P is such that the line PF2F3 is equilateral or something.Wait, let me think differently. The area is (1/2)| -x - y +1 |. So, to maximize this, we need to maximize | -x - y +1 |.But given the constraint of the curve C, which is the sum of distances to F1, F2, F3 is 2√2.I might need to use some optimization techniques here.Let me denote S = |PF1| + |PF2| + |PF3| = 2√2.We need to maximize | -x - y +1 |.Alternatively, perhaps we can use the method of Lagrange multipliers.Define the function to maximize: f(x,y) = | -x - y +1 |.Subject to the constraint g(x,y) = √[(x+1)^2 + y^2] + √[(x-1)^2 + y^2] + √[x^2 + (y -1)^2] - 2√2 = 0.This is a constrained optimization problem.But this might be complicated. Alternatively, maybe I can consider symmetry or specific points.Wait, let me consider the point P where x = y. Maybe that could give some symmetry.Let me set x = y, then the equation becomes:√[(x +1)^2 + x^2] + √[(x -1)^2 + x^2] + √[x^2 + (x -1)^2] = 2√2.Simplify each term:First term: √[(x +1)^2 + x^2] = √[x^2 + 2x +1 + x^2] = √[2x^2 + 2x +1].Second term: √[(x -1)^2 + x^2] = √[x^2 - 2x +1 + x^2] = √[2x^2 - 2x +1].Third term: √[x^2 + (x -1)^2] = √[x^2 + x^2 - 2x +1] = √[2x^2 - 2x +1].So, the equation becomes:√(2x^2 + 2x +1) + √(2x^2 - 2x +1) + √(2x^2 - 2x +1) = 2√2.Simplify:√(2x^2 + 2x +1) + 2√(2x^2 - 2x +1) = 2√2.Let me denote u = x. Then, the equation is:√(2u^2 + 2u +1) + 2√(2u^2 - 2u +1) = 2√2.This still looks complicated. Maybe try specific values.Let me try u = 0:√(0 +0 +1) + 2√(0 -0 +1) = 1 + 2*1 = 3 ≠ 2√2.u = 1:√(2 + 2 +1) + 2√(2 - 2 +1) = √5 + 2√1 ≈ 2.236 + 2 = 4.236 ≠ 2√2.u = 0.5:√(2*(0.25) + 2*(0.5) +1) + 2√(2*(0.25) - 2*(0.5) +1)= √(0.5 +1 +1) + 2√(0.5 -1 +1)= √2.5 + 2√0.5 ≈ 1.581 + 2*0.707 ≈ 1.581 + 1.414 ≈ 3.0 ≈ 2√2 ≈ 2.828.Not quite, but close.Wait, maybe u = something else.Alternatively, perhaps this approach isn't the best. Maybe I should consider that the maximum area occurs when P is such that the height is maximized, which might be when P is at a certain point relative to F2F3.Alternatively, maybe the maximum area is indeed √3 / 2, but I need to verify.Wait, √3 / 2 is approximately 0.866. Earlier, at (0,0), the area was 1/2 = 0.5. So, maybe there's a point where the area is higher.Alternatively, perhaps the maximum area is achieved when P is at a point where the line PF2F3 is such that P is at a certain height.Wait, maybe I can use the fact that the area is (1/2)| -x - y +1 |, so to maximize this, we need to maximize | -x - y +1 |.Given that P is on curve C, which is the set of points where the sum of distances to F1, F2, F3 is 2√2.I think this might require some calculus or optimization, but since I'm a bit stuck, maybe I can look for another approach.Alternatively, perhaps the maximum area is indeed √3 / 2, but I need to confirm.Wait, let me consider the ellipse with foci at F2 and F3, with major axis length 2√2. Wait, but the sum of distances to F2 and F3 would be 2√2 - |PF1|, which complicates things.Alternatively, maybe the maximum area occurs when P is such that the triangle PF2F3 is equilateral, but that might not necessarily be the case.Alternatively, perhaps I can consider the point P where the derivative of the area with respect to x or y is zero, given the constraint.But this might be too involved for me right now. Given that I'm not sure, maybe I should consider that conclusion ④ is incorrect because the maximum area is actually less than √3 / 2.Wait, earlier, when I considered the point (0,1), the area was zero, and at (0,0), it was 0.5. Maybe the maximum is somewhere in between.Alternatively, perhaps the maximum area is indeed √3 / 2, but I need to verify.Wait, let me compute the area when P is at (1/2, 1/2). Then, the area would be (1/2)| -1/2 -1/2 +1 | = (1/2)| -1 +1 | = 0. So, that's not helpful.Alternatively, maybe P is at (0, something). Let me try P(0, y). Then, the area is (1/2)| -0 - y +1 | = (1/2)|1 - y|.Given that P is on curve C, so:√[(0 +1)^2 + y^2] + √[(0 -1)^2 + y^2] + √[0^2 + (y -1)^2] = 2√2.Which simplifies to:2√(1 + y^2) + |y -1| = 2√2.As before, when y =1, this holds. So, the area is (1/2)|1 -1| = 0.If y <1, say y=0, area is (1/2)|1 -0| = 0.5.If y=0.5, area is (1/2)|1 -0.5| = 0.25.So, the maximum area when x=0 is 0.5.But earlier, when x=0.5, the area was 0.5 as well.Wait, maybe the maximum area is 0.5, but conclusion ④ says it's √3 / 2 ≈ 0.866, which is higher.Alternatively, maybe I made a mistake in my earlier reasoning.Wait, let me think again. The area is (1/2)| -x - y +1 |. To maximize this, we need to maximize | -x - y +1 |.Given that P is on curve C, which is the set of points where the sum of distances to F1, F2, F3 is 2√2.I think the maximum value of | -x - y +1 | occurs when P is as far as possible from the line F2F3, which is y = -x +1.But how far can P be from this line?Alternatively, maybe the maximum occurs when P is at a certain point where the gradient of the area function is parallel to the gradient of the constraint function.This would require using Lagrange multipliers.Let me set up the Lagrangian:L(x,y,λ) = (-x - y +1) + λ(√[(x+1)^2 + y^2] + √[(x-1)^2 + y^2] + √[x^2 + (y -1)^2] - 2√2).Wait, actually, since we are maximizing | -x - y +1 |, and the maximum occurs when -x - y +1 is positive or negative. Let's assume it's positive, so we can drop the absolute value.So, we need to maximize f(x,y) = -x - y +1, subject to g(x,y) = √[(x+1)^2 + y^2] + √[(x-1)^2 + y^2] + √[x^2 + (y -1)^2] - 2√2 = 0.The Lagrangian is:L = -x - y +1 + λ(√[(x+1)^2 + y^2] + √[(x-1)^2 + y^2] + √[x^2 + (y -1)^2] - 2√2).Taking partial derivatives:∂L/∂x = -1 + λ*( (x +1)/√[(x+1)^2 + y^2] + (x -1)/√[(x-1)^2 + y^2] + x/√[x^2 + (y -1)^2] ) = 0.∂L/∂y = -1 + λ*( y/√[(x+1)^2 + y^2] + y/√[(x-1)^2 + y^2] + (y -1)/√[x^2 + (y -1)^2] ) = 0.∂L/∂λ = √[(x+1)^2 + y^2] + √[(x-1)^2 + y^2] + √[x^2 + (y -1)^2] - 2√2 = 0.This system of equations is quite complicated. Maybe I can look for symmetry or specific points where the derivatives are zero.Alternatively, perhaps the maximum occurs when P is at a point where the line PF2F3 is such that the height is maximized.Alternatively, maybe the maximum area is indeed √3 / 2, but I'm not sure.Given that I'm stuck, maybe I should consider that conclusion ④ is incorrect because the maximum area is actually less than √3 / 2.Alternatively, perhaps the maximum area is indeed √3 / 2, but I need to confirm.Wait, let me consider the point P where x = y. Earlier, when x = y = 0.5, the area was 0.5. Maybe if I take x = y = something else, I can get a higher area.Alternatively, perhaps the maximum area is achieved when P is at a certain point where the derivative is zero, but I can't compute it right now.Given that I'm not sure, maybe I should consider that conclusion ④ is incorrect because the maximum area is actually less than √3 / 2.Alternatively, perhaps the maximum area is indeed √3 / 2, but I need to confirm.Wait, let me think about the ellipse with foci at F2 and F3, with major axis length 2√2. The maximum area of triangle PF2F3 would be when P is at the top of the ellipse, which is at a distance of √2 from both F2 and F3.Wait, but the sum of distances to F2 and F3 would be 2√2, but we also have the distance to F1, which complicates things.Alternatively, maybe the maximum area occurs when P is at a certain point, but I'm not sure.Given that I'm stuck, I think I'll have to consider that conclusion ④ is incorrect because the maximum area is actually less than √3 / 2.So, summarizing:Conclusion ①: Incorrect.Conclusion ②: Incorrect.Conclusion ③: Correct.Conclusion ④: Incorrect.Therefore, only conclusion ③ is correct.