💡Okay, so I have these two equations to solve. Let me take them one at a time.Starting with the first equation: ( x^{2} - 4x - 1 = 0 ). Hmm, this looks like a quadratic equation. I remember that quadratic equations can be solved using factoring, completing the square, or the quadratic formula. Let me see if this can be factored easily.Looking at ( x^{2} - 4x - 1 ), I need two numbers that multiply to -1 and add up to -4. Hmm, that doesn't seem straightforward. Maybe factoring isn't the best approach here. Let me try completing the square instead.To complete the square, I'll rewrite the equation as ( x^{2} - 4x = 1 ). Now, I need to add a constant to both sides to make the left side a perfect square trinomial. The coefficient of x is -4, so I take half of that, which is -2, and square it to get 4. Adding 4 to both sides gives me:( x^{2} - 4x + 4 = 1 + 4 )Simplifying both sides:( (x - 2)^{2} = 5 )Now, to solve for x, I take the square root of both sides:( x - 2 = pm sqrt{5} )So, adding 2 to both sides gives me the solutions:( x = 2 + sqrt{5} ) and ( x = 2 - sqrt{5} )Alright, that seems good for the first equation.Now, moving on to the second equation: ( (x + 4)^{2} = 5(x + 4) ). This looks a bit more complicated, but maybe I can simplify it.First, I'll expand the left side:( (x + 4)^{2} = x^{2} + 8x + 16 )So, the equation becomes:( x^{2} + 8x + 16 = 5x + 20 )Now, I'll move all terms to one side to set the equation to zero:( x^{2} + 8x + 16 - 5x - 20 = 0 )Simplifying like terms:( x^{2} + 3x - 4 = 0 )Okay, now I have a quadratic equation again. Let me try factoring this one. I need two numbers that multiply to -4 and add up to 3. Hmm, 4 and -1 work because 4 * (-1) = -4 and 4 + (-1) = 3.So, factoring the quadratic:( (x + 4)(x - 1) = 0 )Setting each factor equal to zero gives me the solutions:( x + 4 = 0 ) ⇒ ( x = -4 )and( x - 1 = 0 ) ⇒ ( x = 1 )Wait a minute, I think I could have solved the second equation without expanding. Let me try that approach again to double-check.Starting with ( (x + 4)^{2} = 5(x + 4) ), I can subtract ( 5(x + 4) ) from both sides:( (x + 4)^{2} - 5(x + 4) = 0 )Now, factor out ( (x + 4) ):( (x + 4)(x + 4 - 5) = 0 )Simplifying inside the second factor:( (x + 4)(x - 1) = 0 )So, the solutions are ( x = -4 ) and ( x = 1 ), which matches what I got earlier. That's reassuring.I think I've got both equations solved correctly. The first equation has solutions involving square roots, and the second one has integer solutions. Let me just write them out clearly.For the first equation, ( x^{2} - 4x - 1 = 0 ), the solutions are ( x = 2 + sqrt{5} ) and ( x = 2 - sqrt{5} ).For the second equation, ( (x + 4)^{2} = 5(x + 4) ), the solutions are ( x = -4 ) and ( x = 1 ).I don't see any mistakes in my calculations, and both methods for the second equation gave me the same results. I think I'm confident with these solutions.

💡Okay, so I have this probability problem about car owners purchasing insurance types A and B. Let me try to understand what it's asking and figure out how to solve it step by step.First, part (Ⅰ) asks for the probability that one car owner purchases at least one of the two types of insurance, A or B. I know that probabilities can sometimes be tricky, especially when dealing with "at least one" scenarios. I remember that the probability of at least one event happening can be found using the principle of inclusion-exclusion. The problem states that the probability of purchasing insurance type A is 0.5. That seems straightforward. It also mentions that the probability of purchasing insurance type B but not type A is 0.3. Hmm, so that means if someone buys B, they might also buy A, but in this case, it's specifically those who buy B and not A. I think I need to find the total probability of someone buying A or B or both. To do that, I should probably find the probability of buying B first. Let me denote the probability of buying B as P(B). Since buying B but not A is 0.3, and buying A is 0.5, I can set up an equation. If P(B) is the total probability of buying B, then the probability of buying B but not A is P(B) minus the probability of buying both A and B. So, P(B but not A) = P(B) - P(A and B). But I don't know P(A and B) yet. Wait, maybe there's another way. Since the problem says that buying B but not A is 0.3, and buying A is 0.5, perhaps I can find P(B) directly. If I assume that buying A and B are independent events, which the problem does say, then P(A and B) = P(A) * P(B). But hold on, the problem says "each car owner's insurance purchase is independent," which I think means that the purchase of A and B are independent events. So, if that's the case, then P(A and B) = P(A) * P(B). Given that, and knowing that P(B but not A) = P(B) - P(A and B) = 0.3, I can substitute P(A and B) with P(A) * P(B). So, 0.3 = P(B) - 0.5 * P(B). Simplifying that, 0.3 = P(B) * (1 - 0.5) = P(B) * 0.5. Therefore, P(B) = 0.3 / 0.5 = 0.6.Okay, so the probability of buying B is 0.6. Now, to find the probability of buying at least one of A or B, which is P(A or B), I can use the inclusion-exclusion principle: P(A or B) = P(A) + P(B) - P(A and B). We already know P(A) = 0.5, P(B) = 0.6, and P(A and B) = 0.5 * 0.6 = 0.3. So, plugging those in: P(A or B) = 0.5 + 0.6 - 0.3 = 0.8.So, the probability that a car owner buys at least one of the two insurances is 0.8.Now, moving on to part (Ⅱ). It says that X represents the number of car owners out of 100 who do not purchase either insurance type A or B. We need to find the expected value of X.First, I need to find the probability that a single car owner does not purchase either insurance. From part (Ⅰ), we found that the probability of purchasing at least one insurance is 0.8. Therefore, the probability of not purchasing either is 1 - 0.8 = 0.2.So, each car owner has a 0.2 chance of not purchasing either insurance. Since we're dealing with 100 car owners, and each is independent, this is a binomial distribution scenario. The expected value for a binomial distribution is n * p, where n is the number of trials and p is the probability of success. In this case, "success" is not purchasing either insurance, so p = 0.2, and n = 100.Therefore, the expected value E(X) = 100 * 0.2 = 20.Wait, let me double-check that. If each car owner has a 0.2 chance of not buying either insurance, then on average, out of 100 car owners, 20 would not buy either. That seems reasonable.So, summarizing:Ⅰ. The probability of purchasing at least one insurance is 0.8.Ⅱ. The expected number of car owners not purchasing either insurance is 20.I think that makes sense. I don't see any mistakes in my calculations, but let me just go through it again quickly.For part (Ⅰ):- P(A) = 0.5- P(B but not A) = 0.3Since P(B but not A) = P(B) - P(A and B), and assuming independence, P(A and B) = P(A) * P(B). So, 0.3 = P(B) - 0.5 * P(B) => 0.3 = 0.5 * P(B) => P(B) = 0.6.Then, P(A or B) = 0.5 + 0.6 - 0.3 = 0.8. That checks out.For part (Ⅱ):- Probability of not buying either = 1 - 0.8 = 0.2- Expected value for 100 car owners = 100 * 0.2 = 20. That also seems correct.Alright, I feel confident with these answers.

💡Okay, so I have this problem here about vectors and cross products. Let me try to figure it out step by step. The vectors given are m = (1, 0) and n = (-1, √3). I need to find the magnitude of their cross product, |m × n|.First, I remember that the cross product of two vectors in two dimensions can be calculated using the determinant formula. The cross product in two dimensions is a scalar, but since we're talking about the magnitude, it should be straightforward.Wait, actually, cross product is typically defined for three-dimensional vectors, right? But since these are two-dimensional vectors, maybe they're embedded in three dimensions with the z-component being zero. So, m would be (1, 0, 0) and n would be (-1, √3, 0). That makes sense because the cross product in three dimensions will give us a vector perpendicular to both, which would be along the z-axis.So, the cross product formula for three-dimensional vectors is:m × n = |i   j   k|         |1   0   0|         |-1 √3  0|Calculating this determinant, we get:i*(0*0 - 0*√3) - j*(1*0 - 0*(-1)) + k*(1*√3 - 0*(-1))Simplifying each component:i*(0 - 0) - j*(0 - 0) + k*(√3 - 0)So, that simplifies to:0i - 0j + √3 kTherefore, the cross product vector is (0, 0, √3). The magnitude of this vector is just the absolute value of the z-component, which is √3.Alternatively, I remember another formula for the magnitude of the cross product: |m||n|sinθ, where θ is the angle between the vectors. Let me try that method too to verify.First, find the magnitudes of m and n.|m| = √(1² + 0²) = √1 = 1|n| = √((-1)² + (√3)²) = √(1 + 3) = √4 = 2Next, find the angle θ between m and n. I can use the dot product formula to find cosθ.m · n = (1)(-1) + (0)(√3) = -1 + 0 = -1So, cosθ = (m · n)/(|m||n|) = (-1)/(1*2) = -1/2Therefore, θ = arccos(-1/2). I know that cos(120°) = -1/2, so θ = 120°.Now, sinθ = sin(120°) = √3/2So, |m × n| = |m||n|sinθ = 1*2*(√3/2) = √3That matches the result from the determinant method. So, the magnitude of the cross product is √3.Looking back at the options, D is √3, so that must be the correct answer.I think I've got it. The key was remembering how to compute the cross product for two-dimensional vectors by treating them as three-dimensional with zero z-components. Then, using either the determinant method or the magnitude-angle formula gives the same result. It's reassuring that both methods lead to the same answer, so I'm confident that D is correct.**Final Answer**boxed{D}

💡Okay, so I have this sequence problem here. It says that in the sequence {a_n}, a_2 is 3, a_3 is 6, and there's a recursive relation: a_{n+2} = a_{n+1} - 1/a_n. I need to find the value of a_5. The options are A. 12, B. 24, C. 11/4, D. 11/2.Hmm, let me try to figure this out step by step. First, I know that a_2 is 3 and a_3 is 6. The recursive formula is a_{n+2} = a_{n+1} - 1/a_n. So, to find a_5, I probably need to find a_4 first, and then use that to find a_5.Let me write down the formula again for clarity: a_{n+2} = a_{n+1} - 1/a_n. So, if I plug in n=2, then a_{4} = a_{3} - 1/a_2. Since a_3 is 6 and a_2 is 3, that would be a_4 = 6 - 1/3. Let me calculate that: 6 is 18/3, so 18/3 - 1/3 is 17/3. So, a_4 is 17/3.Now, to find a_5, I can use the same formula. Let me plug in n=3: a_{5} = a_{4} - 1/a_3. We just found that a_4 is 17/3, and a_3 is 6. So, a_5 = 17/3 - 1/6. Hmm, I need to subtract these two fractions. To do that, they need a common denominator. The denominators are 3 and 6, so the common denominator is 6.Let me convert 17/3 to sixths: 17/3 is equal to (17*2)/(3*2) = 34/6. So, now I have 34/6 - 1/6 = (34 - 1)/6 = 33/6. Simplifying 33/6, I can divide numerator and denominator by 3: 33 ÷ 3 = 11 and 6 ÷ 3 = 2, so it's 11/2.Wait, so a_5 is 11/2? Let me check my steps again to make sure I didn't make a mistake.First, a_4 = a_3 - 1/a_2 = 6 - 1/3 = 17/3. That seems right.Then, a_5 = a_4 - 1/a_3 = 17/3 - 1/6. Converting 17/3 to sixths: 17*2=34, so 34/6. Then, 34/6 - 1/6 = 33/6 = 11/2. Yeah, that seems correct.Looking back at the options, D is 11/2, so that must be the answer. I don't think I made any calculation errors here. The steps are straightforward: use the recursive formula step by step, calculate each term one by one, and simplify the fractions properly.Just to be thorough, let me write down all the terms:a_2 = 3a_3 = 6a_4 = a_3 - 1/a_2 = 6 - 1/3 = 17/3 ≈ 5.666...a_5 = a_4 - 1/a_3 = 17/3 - 1/6 = 34/6 - 1/6 = 33/6 = 11/2 = 5.5So, a_5 is 11/2, which is 5.5. That makes sense because each time we're subtracting a smaller fraction, so the sequence is decreasing but not by much each time.I think I'm confident with this answer. It's not one of the whole numbers like 12 or 24, so options A and B are out. Between C and D, 11/4 is 2.75 and 11/2 is 5.5. Since our calculation gave us 5.5, which is 11/2, D is the correct choice.**Final Answer**boxed{D}

💡Okay, so Sasha wants to plant trees to reduce the local temperature. I heard that planting trees can help lower temperatures, which is pretty cool. But I need to figure out how much it would cost her to plant enough trees to drop the temperature from 80 degrees to 78.2 degrees. First, I should figure out how much the temperature actually dropped. So, if it went from 80 to 78.2, that means it dropped by 1.8 degrees. I can calculate that by subtracting 78.2 from 80, which gives me 1.8 degrees. Now, I know that each tree planted reduces the temperature by 0.1 degrees. So, to find out how many trees Sasha needs to plant to achieve a 1.8-degree drop, I can divide the total temperature drop by the temperature reduction per tree. That would be 1.8 divided by 0.1, which equals 18 trees. Okay, so Sasha needs to plant 18 trees. Each tree costs 6 to plant. To find the total cost, I should multiply the number of trees by the cost per tree. So, 18 trees times 6 per tree equals 108. Wait, let me double-check that. If each tree costs 6 and she plants 18 trees, then 18 times 6 is indeed 108. Yeah, that makes sense. So, the total cost to plant the trees would be 108. I think that's it. She needs to spend 108 to plant enough trees to reduce the temperature by 1.8 degrees.

💡Okay, so I have this problem here: The function ( g ) is linear and satisfies ( g(k+1) - g(k) = 5 ) for all real numbers ( k ). I need to find ( g(4) - g(7) ). Hmm, let me think about how to approach this.First, I remember that a linear function has the form ( g(k) = mk + b ), where ( m ) is the slope and ( b ) is the y-intercept. Since it's linear, the difference ( g(k+1) - g(k) ) should be constant, right? And in this case, that constant difference is 5. So, that must mean that the slope ( m ) is 5. Let me write that down:( g(k) = 5k + b )Okay, so now I know the general form of the function. But I don't know the value of ( b ). However, I don't think I need ( b ) to find the difference ( g(4) - g(7) ). Let me check.If I compute ( g(4) ), it would be ( 5*4 + b = 20 + b ). Similarly, ( g(7) ) would be ( 5*7 + b = 35 + b ). So, subtracting these, ( g(4) - g(7) = (20 + b) - (35 + b) ). The ( b ) terms cancel out, so that leaves me with ( 20 - 35 = -15 ). Wait, that seems straightforward. But let me make sure I'm not missing anything. The problem says ( g(k+1) - g(k) = 5 ) for all real numbers ( k ), which aligns with the slope being 5. So, the function increases by 5 for each increment in ( k ). Alternatively, maybe I can think about it step by step. If I go from ( k = 4 ) to ( k = 5 ), the function increases by 5. Then from ( k = 5 ) to ( k = 6 ), it increases another 5, and from ( k = 6 ) to ( k = 7 ), another 5. So, from 4 to 7, that's three steps, each increasing by 5, so total increase is 15. Therefore, ( g(7) = g(4) + 15 ). So, ( g(4) - g(7) = g(4) - (g(4) + 15) = -15 ).Yeah, that seems consistent. Both methods give me the same answer. So, I think I can confidently say that ( g(4) - g(7) = -15 ).**Final Answer**The value of ( g(4) - g(7) ) is boxed{-15}.