answer：Okay, so I have this problem where a sociologist is modeling emotional recovery after trauma using a quadratic equation. The equation given is R(t) = -2t² + 12t + 20. There are two questions to answer: first, when does the maximum recovery happen and what is that maximum level? Second, after how many months does the recovery return to the initial level.Alright, let me start with the first question. I remember that quadratic equations graph as parabolas. Since the coefficient of t² is negative (-2), the parabola opens downward. That means the vertex of the parabola is the highest point, which in this context would be the maximum level of emotional recovery. So, to find the time t at which this maximum occurs, I need to find the vertex of the parabola.The general form of a quadratic equation is R(t) = at² + bt + c. In this case, a = -2, b = 12, and c = 20. I recall that the time t at which the vertex occurs is given by the formula t = -b/(2a). Let me plug in the values here.So, t = -12/(2*(-2)) = -12/(-4) = 3. So, the maximum recovery occurs at t = 3 months. Now, to find the maximum level of recovery, I need to substitute t = 3 back into the equation R(t).Calculating R(3): R(3) = -2*(3)² + 12*(3) + 20. Let me compute each term step by step.First, (3)² is 9. Multiply that by -2: -2*9 = -18.Next, 12*3 is 36.Then, we have the constant term 20.So, adding them all together: -18 + 36 + 20. Let's compute that.-18 + 36 is 18, and 18 + 20 is 38. So, R(3) = 38. Therefore, the maximum level of emotional recovery is 38 at 3 months.Wait, let me double-check my calculations to make sure I didn't make a mistake. So, R(3) = -2*(9) + 36 + 20. That's -18 + 36 + 20. Yes, that adds up to 38. Okay, that seems correct.So, the first part is done. The maximum recovery is at t = 3 months, and the level is 38.Now, moving on to the second question: After how many months will the level of emotional recovery return to the initial level experienced immediately after the trauma?Hmm, the initial level is right after the trauma, which would be at t = 0. So, let me find R(0) first.R(0) = -2*(0)² + 12*(0) + 20 = 0 + 0 + 20 = 20. So, the initial level is 20.We need to find the time t when R(t) = 20 again. So, we set up the equation:-2t² + 12t + 20 = 20Subtracting 20 from both sides gives:-2t² + 12t = 0Factor out a common term, which is -2t:-2t(t - 6) = 0So, setting each factor equal to zero:-2t = 0 => t = 0t - 6 = 0 => t = 6So, the solutions are t = 0 and t = 6. Since we're looking for the time after the trauma when the recovery returns to the initial level, t = 0 is the starting point, so the other solution is t = 6 months.Therefore, after 6 months, the emotional recovery level returns to the initial level of 20.Wait, let me just verify that. If I plug t = 6 into R(t):R(6) = -2*(6)² + 12*(6) + 20Compute each term:6² is 36, multiplied by -2 is -72.12*6 is 72.Adding them up: -72 + 72 + 20 = 20. Yep, that's correct.So, the level returns to 20 at t = 6 months.Let me recap:1. The maximum recovery occurs at t = 3 months, with a level of 38.2. The recovery returns to the initial level of 20 at t = 6 months.I think that's all. I don't see any mistakes in my calculations, so I feel confident about these answers.**Final Answer**1. The maximum level of emotional recovery is achieved at boxed{3} months, with a recovery level of boxed{38}.2. The emotional recovery returns to the initial level after boxed{6} months.

answer：First, I need to determine the year(s) when the voter turnout was exactly 70%. To do this, I'll set the equation ( V(t) = -2t^2 + 12t + 60 ) equal to 70 and solve for ( t ).So, I'll start by setting up the equation:[-2t^2 + 12t + 60 = 70]Next, I'll subtract 70 from both sides to set the equation to zero:[-2t^2 + 12t - 10 = 0]To simplify, I'll divide the entire equation by -2:[t^2 - 6t + 5 = 0]Now, I'll factor the quadratic equation:[(t - 1)(t - 5) = 0]Setting each factor equal to zero gives the solutions:[t = 1 quad text{and} quad t = 5]This means the voter turnout was exactly 70% in the 1st and 5th years after the first election.For the second part, I need to find the maximum voter turnout percentage and the corresponding number of years. Since the quadratic equation opens downward (the coefficient of ( t^2 ) is negative), the vertex will give the maximum value.The general form of a quadratic equation is ( at^2 + bt + c ), and the vertex occurs at ( t = -frac{b}{2a} ). Here, ( a = -2 ) and ( b = 12 ), so:[t = -frac{12}{2 times (-2)} = 3]To find the maximum voter turnout, I'll substitute ( t = 3 ) back into the original equation:[V(3) = -2(3)^2 + 12(3) + 60 = -18 + 36 + 60 = 78]Therefore, the maximum voter turnout percentage is 78%, which occurs 3 years after the first election.

answer：Okay, so I've got two problems to solve here. Let me start with the first one.**Problem 1: Formulating and Solving the System of Equations**Alright, Alex is renting skis and a snowboard. The rental shop charges 30 per hour for skis and 50 per hour for a snowboard. He spends a total of 6 hours on the slopes and his total cost is 240. I need to find out how many hours he rented each.Let me define the variables first. Let ( x ) be the number of hours he rented the skis, and ( y ) be the number of hours he rented the snowboard.So, since he spent a total of 6 hours, that gives me the first equation:( x + y = 6 )That's straightforward. Now, for the cost. Skis cost 30 per hour, so the cost for skis is ( 30x ). Similarly, the snowboard costs 50 per hour, so the cost for the snowboard is ( 50y ). The total cost is 240, so:( 30x + 50y = 240 )So now I have a system of two equations:1. ( x + y = 6 )2. ( 30x + 50y = 240 )I need to solve this system. Let me use substitution or elimination. Maybe elimination is easier here.First, let me simplify the second equation. I can divide both sides by 10 to make the numbers smaller:( 3x + 5y = 24 )Now, from the first equation, I can express ( x ) in terms of ( y ):( x = 6 - y )Now, substitute this into the second equation:( 3(6 - y) + 5y = 24 )Let me expand that:( 18 - 3y + 5y = 24 )Combine like terms:( 18 + 2y = 24 )Subtract 18 from both sides:( 2y = 6 )Divide by 2:( y = 3 )So, ( y = 3 ) hours. Then, ( x = 6 - y = 6 - 3 = 3 ) hours.Wait, so he rented both skis and snowboard for 3 hours each? Let me check if that makes sense.Cost for skis: 3 hours * 30/hour = 90Cost for snowboard: 3 hours * 50/hour = 150Total cost: 90 + 150 = 240. Yep, that's correct.So, Alex rented the skis for 3 hours and the snowboard for 3 hours.**Problem 2: Finding the Equation of the Parabola**Alex skied down a slope that's approximately parabolic. The path is given by ( h(x) = -ax^2 + bx + c ). The start and end points are at the same height, which is 0 meters at ( x = 0 ) and ( x = 200 ). The top of the hill is at 100 meters.So, let's parse this.First, the parabola passes through three points: (0, 0), (200, 0), and the vertex is at the maximum height, which is 100 meters. Wait, but the vertex isn't necessarily at the midpoint unless it's symmetric. Hmm.Wait, the problem says he starts at the top of the hill, which is at a height of 100 meters from the baseline. So, the starting point is at ( x = 0 ), ( h(0) = 100 ). But wait, the problem also says the start and end points are at the same height, which is 0 meters. Wait, that seems conflicting.Wait, hold on. Let me read that again."Assume the parabolic path can be represented by the quadratic function ( h(x) = -ax^2 + bx + c ), where ( h(x) ) is the height in meters at a horizontal distance ( x ) meters from the starting point. Given that the start and end points of the path are at the same height (0 meters at ( x = 0 ) and ( x = 200 ))..."Wait, so at ( x = 0 ), ( h(0) = 0 ), and at ( x = 200 ), ( h(200) = 0 ). But the starting point is at a height of 100 meters. Hmm, that seems contradictory.Wait, maybe I misread. Let me check:"Alex estimates that the slope he skied down was approximately parabolic in shape. He starts at the top of the hill, which is at a height of 100 meters from the baseline, and the hill extends 200 meters horizontally."So, he starts at (0, 100) and ends at (200, 0). But the problem says the start and end points are at the same height, which is 0 meters. Wait, that doesn't make sense because he starts at 100 meters and ends at 0 meters.Wait, maybe the problem is saying that the start and end points are at the same height, but that height is 0. So, he starts at (0, 0) and ends at (200, 0), but the top is at 100 meters somewhere in between.Wait, that seems more consistent. So, the parabola goes from (0, 0) to (200, 0), with a maximum height of 100 meters somewhere in between.So, the function is ( h(x) = -ax^2 + bx + c ). Let's plug in the known points.First, at ( x = 0 ), ( h(0) = 0 ):( h(0) = -a(0)^2 + b(0) + c = c = 0 )So, ( c = 0 ). Therefore, the equation simplifies to:( h(x) = -ax^2 + bx )Next, at ( x = 200 ), ( h(200) = 0 ):( h(200) = -a(200)^2 + b(200) = 0 )So:( -40000a + 200b = 0 )Let me write that as:( 200b = 40000a )Divide both sides by 200:( b = 200a )So, ( b = 200a ). So, now, the equation becomes:( h(x) = -ax^2 + 200a x )Now, we also know that the maximum height is 100 meters. Since it's a parabola opening downward (because of the negative coefficient on ( x^2 )), the vertex is the maximum point.The vertex of a parabola ( h(x) = ax^2 + bx + c ) is at ( x = -frac{b}{2a} ). Wait, but in our case, the equation is ( h(x) = -ax^2 + bx ), so the coefficient of ( x^2 ) is ( -a ), and the coefficient of ( x ) is ( b ).So, the x-coordinate of the vertex is:( x = -frac{b}{2(-a)} = frac{b}{2a} )But we have ( b = 200a ), so:( x = frac{200a}{2a} = 100 )So, the vertex is at ( x = 100 ) meters. That makes sense because it's halfway between 0 and 200, so the maximum height occurs at the midpoint.Now, the height at ( x = 100 ) is 100 meters. So, let's plug that into the equation:( h(100) = -a(100)^2 + 200a(100) = 100 )Compute this:( -10000a + 20000a = 100 )Simplify:( 10000a = 100 )Divide both sides by 10000:( a = frac{100}{10000} = frac{1}{100} = 0.01 )So, ( a = 0.01 ). Then, since ( b = 200a ), we have:( b = 200 * 0.01 = 2 )Therefore, the equation is:( h(x) = -0.01x^2 + 2x )Let me write that in a more standard form:( h(x) = -0.01x^2 + 2x )Alternatively, to make it look cleaner, we can write it as:( h(x) = -frac{1}{100}x^2 + 2x )Let me verify this equation with the given points.At ( x = 0 ):( h(0) = 0 + 0 = 0 ). Correct.At ( x = 200 ):( h(200) = -0.01*(200)^2 + 2*200 = -0.01*40000 + 400 = -400 + 400 = 0 ). Correct.At ( x = 100 ):( h(100) = -0.01*(100)^2 + 2*100 = -100 + 200 = 100 ). Correct.So, that seems to satisfy all the conditions.**Final Answer**1. Alex rented the skis for boxed{3} hours and the snowboard for boxed{3} hours.2. The equation of the parabola is boxed{h(x) = -frac{1}{100}x^2 + 2x}.

answer：First, I'll define the width of the vineyard as ( w ) meters. According to the problem, the length is 10 meters more than twice the width, so the length can be expressed as ( 2w + 10 ) meters.Next, I'll set up the equation for the area of the vineyard. The area of a rectangle is given by the product of its length and width, so:[w times (2w + 10) = 200]Expanding this equation, I get:[2w^2 + 10w = 200]To form a standard quadratic equation, I'll subtract 200 from both sides:[2w^2 + 10w - 200 = 0]I can simplify this equation by dividing all terms by 2:[w^2 + 5w - 100 = 0]Now, I'll solve this quadratic equation using the quadratic formula:[w = frac{-b pm sqrt{b^2 - 4ac}}{2a}]Here, ( a = 1 ), ( b = 5 ), and ( c = -100 ). Plugging these values in:[w = frac{-5 pm sqrt{25 + 400}}{2} = frac{-5 pm sqrt{425}}{2}]Since width cannot be negative, I'll take the positive solution:[w = frac{-5 + sqrt{425}}{2} approx 7.53 text{ meters}]The length is then:[2w + 10 approx 2 times 7.53 + 10 approx 25.06 text{ meters}]For the second part, to double the area while keeping the width the same, the new area will be ( 400 ) square meters. Using the same width:[400 = 7.53 times text{new length}]Solving for the new length:[text{new length} = frac{400}{7.53} approx 53.12 text{ meters}]

answer：Alright, so I've got these two math problems to solve, and I need to figure them out step by step. Let's start with the first one.**Problem 1: Solving the Polynomial Equation**The equation given is ( x^3 - 6x^2 + 11x - 6 = 0 ). Hmm, okay. I remember that solving cubic equations can sometimes be tricky, but maybe I can factor this one. Let me think about possible rational roots. The Rational Root Theorem says that any possible rational root, p/q, is a factor of the constant term divided by a factor of the leading coefficient. Here, the constant term is -6, and the leading coefficient is 1. So possible roots are ±1, ±2, ±3, ±6.Let me test these one by one. I'll start with x=1:( 1^3 - 6*1^2 + 11*1 - 6 = 1 - 6 + 11 - 6 = 0 ). Oh, that works! So x=1 is a root.Now, since x=1 is a root, I can factor (x - 1) out of the polynomial. Let's perform polynomial division or use synthetic division. I think synthetic division might be quicker.Setting up synthetic division with root 1:1 | 1 -6 11 -6Bring down the 1.Multiply 1 by 1: 1. Add to next coefficient: -6 + 1 = -5.Multiply 1 by -5: -5. Add to next coefficient: 11 + (-5) = 6.Multiply 1 by 6: 6. Add to last coefficient: -6 + 6 = 0. Perfect, no remainder.So after factoring out (x - 1), we're left with ( x^2 - 5x + 6 ). Now, let's factor this quadratic.Looking for two numbers that multiply to 6 and add to -5. Those would be -2 and -3.So, ( x^2 - 5x + 6 = (x - 2)(x - 3) ).Therefore, the original polynomial factors to ( (x - 1)(x - 2)(x - 3) ). So the roots are x=1, x=2, and x=3.Wait, let me double-check by plugging in x=2:( 2^3 - 6*2^2 + 11*2 - 6 = 8 - 24 + 22 - 6 = 0 ). Yep, works.And x=3:( 3^3 - 6*3^2 + 11*3 - 6 = 27 - 54 + 33 - 6 = 0 ). Also works.Alright, so that seems solid.**Problem 2: Solving the Rational Equation**The equation is ( frac{3x + 7}{x - 2} = frac{x + 5}{x + 1} ). Okay, so we have a proportion here. To solve for x, I can cross-multiply to eliminate the denominators. But before I do that, I should note the restrictions on x. The denominators cannot be zero, so x ≠ 2 and x ≠ -1.Cross-multiplying gives:( (3x + 7)(x + 1) = (x + 5)(x - 2) ).Let me expand both sides.Left side: ( 3x(x) + 3x(1) + 7(x) + 7(1) = 3x^2 + 3x + 7x + 7 = 3x^2 + 10x + 7 ).Right side: ( x(x) + x(-2) + 5(x) + 5(-2) = x^2 - 2x + 5x - 10 = x^2 + 3x - 10 ).So now, the equation becomes:( 3x^2 + 10x + 7 = x^2 + 3x - 10 ).Let me bring all terms to the left side:( 3x^2 + 10x + 7 - x^2 - 3x + 10 = 0 ).Simplify:( 2x^2 + 7x + 17 = 0 ).Wait, that seems a bit complicated. Let me check my expansion again to make sure I didn't make a mistake.Left side: (3x + 7)(x + 1):First term: 3x * x = 3x²Outer: 3x * 1 = 3xInner: 7 * x = 7xLast: 7 * 1 = 7So total: 3x² + 3x + 7x + 7 = 3x² + 10x + 7. That seems correct.Right side: (x + 5)(x - 2):First: x * x = x²Outer: x * (-2) = -2xInner: 5 * x = 5xLast: 5 * (-2) = -10So total: x² - 2x + 5x -10 = x² + 3x -10. That also seems correct.So subtracting the right side from both sides:3x² + 10x + 7 - x² - 3x + 10 = 0Simplify:(3x² - x²) + (10x - 3x) + (7 + 10) = 02x² + 7x + 17 = 0Hmm, okay. So quadratic equation: 2x² +7x +17=0.Let me compute the discriminant to see if there are real solutions.Discriminant D = b² - 4ac = 7² - 4*2*17 = 49 - 136 = -87.Negative discriminant, so no real solutions. That means the equation has no solution in real numbers.Wait, but let me think again. Did I cross-multiply correctly? Because sometimes when you cross-multiply, you might introduce extraneous solutions or sometimes lose solutions, but in this case, since we ended up with a quadratic with no real roots, it suggests that the original equation has no solution.But let me verify by plugging in some numbers.Suppose x=0:Left side: (0 +7)/(0 -2) = 7/-2 = -3.5Right side: (0 +5)/(0 +1) = 5/1 = 5Not equal.x=1:Left: (3 +7)/(1 -2) = 10/-1 = -10Right: (1 +5)/(1 +1) = 6/2 = 3Not equal.x=3:Left: (9 +7)/(3 -2) = 16/1 =16Right: (3 +5)/(3 +1)=8/4=2Not equal.x=-2:Left: (-6 +7)/(-2 -2)=1/-4=-0.25Right: (-2 +5)/(-2 +1)=3/-1=-3Not equal.So yeah, seems like no solution. So the answer is no real solution.But wait, let me check if I did the cross-multiplication correctly.Original equation:(3x +7)/(x -2) = (x +5)/(x +1)Cross-multiplying: (3x +7)(x +1) = (x +5)(x -2). Yep, that's correct.So, unless I made a mistake in expanding, which I don't think I did, the equation reduces to 2x² +7x +17=0, which has no real roots.Therefore, the equation has no solution.Alternatively, maybe I should present it as no real solution or state that there are no solutions.But in the context of the problem, it's about distributing resources equally, so maybe expecting a real solution? Hmm, but mathematically, it's clear that there's no real solution.So, I think that's the answer.**Final Answer**1. The roots are boxed{1}, boxed{2}, and boxed{3}.2. There is no real solution, so the answer is boxed{text{No solution}}.

answer：Okay, so I have this polynomial equation: f(x) = 2x³ - 7x² + ax - 14. And I know that x = 2 is a root of this equation. I need to find the value of 'a' first. Hmm, let's think about how to approach this.Since x = 2 is a root, that means when I plug in x = 2 into the equation, the result should be zero. So, I can substitute x with 2 and solve for 'a'. Let me write that out:f(2) = 2*(2)³ - 7*(2)² + a*(2) - 14 = 0Calculating each term step by step:First, 2*(2)³. 2³ is 8, so 2*8 is 16.Next, -7*(2)². 2² is 4, so -7*4 is -28.Then, a*(2) is just 2a.Lastly, the constant term is -14.Putting it all together:16 - 28 + 2a - 14 = 0Let me simplify this:16 - 28 is -12.-12 - 14 is -26.So, the equation becomes: -26 + 2a = 0To solve for 'a', I can add 26 to both sides:2a = 26Then, divide both sides by 2:a = 13Okay, so the value of 'a' is 13. That wasn't too bad.Now, moving on to the second part: finding the other roots of the polynomial equation. Since we know that x = 2 is a root, we can factor out (x - 2) from the polynomial. This will leave us with a quadratic equation, which we can then solve to find the other roots.So, the polynomial is f(x) = 2x³ - 7x² + 13x - 14. Let's perform polynomial division or use synthetic division to factor out (x - 2).I think synthetic division might be quicker here. Let me set it up.The coefficients of the polynomial are 2, -7, 13, -14.We're dividing by (x - 2), so we use 2 in synthetic division.Bring down the 2.Multiply 2 by 2, which is 4. Add that to the next coefficient: -7 + 4 = -3.Multiply -3 by 2, which is -6. Add that to the next coefficient: 13 + (-6) = 7.Multiply 7 by 2, which is 14. Add that to the last coefficient: -14 + 14 = 0. Perfect, no remainder.So, the result of the division is 2x² - 3x + 7. Therefore, the polynomial factors as (x - 2)(2x² - 3x + 7).Now, to find the other roots, we need to solve 2x² - 3x + 7 = 0.This is a quadratic equation, so we can use the quadratic formula:x = [3 ± sqrt( (-3)² - 4*2*7 )]/(2*2)Calculating the discriminant first:D = (-3)² - 4*2*7 = 9 - 56 = -47Oh, the discriminant is negative, which means the other roots are complex numbers. That makes sense because the original polynomial has real coefficients, so complex roots come in conjugate pairs.So, the roots are:x = [3 ± sqrt(-47)]/4 = [3 ± i*sqrt(47)]/4Therefore, the other two roots are (3 + i√47)/4 and (3 - i√47)/4.Let me just recap to make sure I didn't make any mistakes. I substituted x = 2 into the equation to find 'a', which gave me a = 13. Then, I used synthetic division to factor out (x - 2), resulting in a quadratic. Solving that quadratic gave me complex roots because the discriminant was negative. That all seems consistent.I think that's it. I don't see any errors in my calculations, so I'm confident with these results.**Final Answer**1. The value of (a) is boxed{13}.2. The other roots of the polynomial are boxed{frac{3 + isqrt{47}}{4}} and boxed{frac{3 - isqrt{47}}{4}}.