answer:To solve the given problem, we will address each part step by step.1. Determine the number of practice hours, ( x ), that will maximize the quality score ( Q(x) ). What is the maximum quality score?The function given is a quadratic function in the form ( Q(x) = -2x^2 + 8x + 10 ), which is a parabola that opens downwards (since the coefficient of ( x^2 ) is negative). The maximum value of a quadratic function ( ax^2 + bx + c ) occurs at the vertex, and the ( x )-coordinate of the vertex can be found using the formula:[x = -frac{b}{2a}]For the given function, ( a = -2 ) and ( b = 8 ). Plugging these into the formula gives:[x = -frac{8}{2(-2)} = -frac{8}{-4} = 2]Thus, the number of practice hours that maximizes the quality score is ( x = 2 ).To find the maximum quality score, we substitute ( x = 2 ) back into the function ( Q(x) ):[Q(2) = -2(2)^2 + 8(2) + 10 = -2(4) + 16 + 10 = -8 + 16 + 10 = 18]So, the maximum quality score is 18.2. Solve the inequality ( -2x^2 + 8x + 10 geq 20 ) to find the range of practice hours needed to achieve a score of at least 20.First, rewrite the inequality:[-2x^2 + 8x + 10 geq 20]Subtract 20 from both sides to set the inequality to zero:[-2x^2 + 8x + 10 - 20 geq 0]Simplify:[-2x^2 + 8x - 10 geq 0]Divide the entire inequality by (-2) (note that this will flip the inequality sign):[x^2 - 4x + 5 leq 0]Now, we solve the equation ( x^2 - 4x + 5 = 0 ) to find critical points. Use the quadratic formula:[x = frac{-b pm sqrt{b^2 - 4ac}}{2a}]Here, ( a = 1 ), ( b = -4 ), and ( c = 5 ). Calculate the discriminant:[b^2 - 4ac = (-4)^2 - 4(1)(5) = 16 - 20 = -4]The discriminant is negative, meaning there are no real roots, and the quadratic ( x^2 - 4x + 5 ) does not intersect the x-axis. Since the parabola opens upwards and is entirely above the x-axis, it is never less than or equal to zero. Therefore, there are no real values of ( x ) such that ( -2x^2 + 8x + 10 geq 20 ).Final Answer: The final answer is that the maximum quality score is 18, achieved with 2 practice hours, and there are no practice hours that achieve a quality score of at least 20 given the constraints. I hope it is correct.

answer:To solve the given problem, we need to find the difference in the number of voters between Party A and Party B, and then solve for the number of election cycles, ( n ), when this difference is exactly 10. Step 1: Determine the Expression for the DifferenceThe number of voters for Party A is given by:[ A(n) = 2n^2 + 3n + 5 ]The number of voters for Party B is given by:[ B(n) = n^2 + 4n + 2 ]The difference in the number of voters supporting Party A and Party B is:[ D(n) = A(n) - B(n) ]Substituting the expressions for ( A(n) ) and ( B(n) ):[D(n) = (2n^2 + 3n + 5) - (n^2 + 4n + 2)]Simplifying the expression:[D(n) = 2n^2 + 3n + 5 - n^2 - 4n - 2]Combine like terms:[D(n) = (2n^2 - n^2) + (3n - 4n) + (5 - 2)][D(n) = n^2 - n + 3] Step 2: Find ( n ) when ( D(n) = 10 )We need to solve the equation:[ n^2 - n + 3 = 10 ]Subtract 10 from both sides to set the equation to zero:[ n^2 - n + 3 - 10 = 0 ][ n^2 - n - 7 = 0 ]This is a quadratic equation of the form ( ax^2 + bx + c = 0 ), where ( a = 1 ), ( b = -1 ), and ( c = -7 ).We can solve this using the quadratic formula:[n = frac{-b pm sqrt{b^2 - 4ac}}{2a}]Substitute the values of ( a ), ( b ), and ( c ):[n = frac{-(-1) pm sqrt{(-1)^2 - 4 cdot 1 cdot (-7)}}{2 cdot 1}][n = frac{1 pm sqrt{1 + 28}}{2}][n = frac{1 pm sqrt{29}}{2}]Since ( n ) represents the number of election cycles, it must be a non-negative integer. The expression (sqrt{29}) is irrational and approximately equal to 5.385. Therefore, both ( frac{1 + sqrt{29}}{2} ) and ( frac{1 - sqrt{29}}{2} ) are not integers, meaning there is no integer solution for ( n ). ConclusionThere is no integer value of ( n ) such that the difference in the number of voters is exactly 10.Final Answer: The final answer is there is no integer ( n ) for which the difference is exactly 10. I hope it is correct.

answer:To solve the given problem, we need to address both parts step-by-step. Part 1: Calculating the Total Production CostThe cost to produce each craft is given by the polynomial ( C(x) = 3x^2 - 5x + 2 ), where ( x ) is the number of crafts. We need to calculate the total production cost for 10 crafts.First, substitute ( x = 10 ) into the polynomial:[C(10) = 3(10)^2 - 5(10) + 2]Calculate each term:- ( 3(10)^2 = 3 times 100 = 300 )- ( -5(10) = -50 )Now, substitute these values back into the equation:[C(10) = 300 - 50 + 2 = 252]Therefore, the total production cost for 10 crafts is 252. Part 2: Solving the Revenue InequalityThe revenue from selling crafts is calculated by multiplying the number of crafts sold by the price per craft. The price per craft is 15. We want the revenue to be at least 150.Let ( y ) represent the number of crafts sold. The revenue ( R(y) ) is given by:[R(y) = 15y]We need to solve the inequality:[15y geq 150]Divide both sides by 15 to solve for ( y ):[y geq frac{150}{15}][y geq 10]Thus, she needs to sell at least 10 crafts to ensure that the revenue is at least 150.---Final Answer: The total production cost is 252, and she needs to sell at least 10 crafts. I hope it is correct.

answer:To solve the given problem, we need to address both parts separately: finding the roots of the polynomial equation and solving the inequality.1. Finding the roots of the polynomial ( f(x) = 2x^3 - 9x^2 + 12x - 4 = 0 ) We need to find the real solutions for this cubic polynomial equation. We will attempt to use the Rational Root Theorem to find any possible rational roots. According to the theorem, any rational root, expressed in lowest terms ( frac{p}{q} ), is such that ( p ) is a factor of the constant term (-4), and ( q ) is a factor of the leading coefficient (2). Possible rational roots are ( pm 1, pm 2, pm 4, pm frac{1}{2}, pm frac{1}{4} ). We will test these values in the polynomial ( f(x) ). Let's start with ( x = 1 ): [ f(1) = 2(1)^3 - 9(1)^2 + 12(1) - 4 = 2 - 9 + 12 - 4 = 1 ] Not a root. Next, ( x = 2 ): [ f(2) = 2(2)^3 - 9(2)^2 + 12(2) - 4 = 16 - 36 + 24 - 4 = 0 ] So, ( x = 2 ) is a root. Since ( x = 2 ) is a root, we can perform synthetic division to divide the polynomial by ( (x - 2) ). [ begin{array}{r|rrrr} 2 & 2 & -9 & 12 & -4 & & 4 & -10 & 4 hline & 2 & -5 & 2 & 0 end{array} ] The quotient is ( 2x^2 - 5x + 2 ). We solve ( 2x^2 - 5x + 2 = 0 ) using the quadratic formula: [ x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ] where ( a = 2 ), ( b = -5 ), ( c = 2 ). [ x = frac{-(-5) pm sqrt{(-5)^2 - 4 cdot 2 cdot 2}}{2 cdot 2} ] [ x = frac{5 pm sqrt{25 - 16}}{4} ] [ x = frac{5 pm sqrt{9}}{4} ] [ x = frac{5 pm 3}{4} ] So the roots are: [ x = frac{8}{4} = 2 quad text{and} quad x = frac{2}{4} = frac{1}{2} ] Therefore, the real solutions for ( f(x) = 0 ) are ( x = 2 ) and ( x = frac{1}{2} ).2. Solving the inequality ( 2x^3 - 9x^2 + 12x - 4 leq 8 ) First, simplify the inequality: [ 2x^3 - 9x^2 + 12x - 4 leq 8 ] [ 2x^3 - 9x^2 + 12x - 12 leq 0 ] We already know that ( f(x) = 2x^3 - 9x^2 + 12x - 4 ), so: [ f(x) - 8 = 2x^3 - 9x^2 + 12x - 12 ] The roots of the polynomial ( 2x^3 - 9x^2 + 12x - 12 = 0 ) are the same as ( f(x) = 0 ) when shifted by 8 units down. Let's find the polynomial roots again by factoring and using the roots of the first equation. The critical points are ( x = 2 ) and ( x = frac{1}{2} ) from the original equation. We check intervals between

answer:To solve the given problem, we will tackle it in two parts as outlined in the problem. Part 1: Determine the number of years for savings to cover installation cost1. Define the variables: - Let ( t ) be the number of years the homeowner plans to stay in the house. - The annual savings on energy costs is 2,500.2. Write the equation for total savings: [ text{Total Savings} = 2500 times t ]3. Set the total savings equal to the installation cost to find ( t ): [ 2500 times t = 20000 ]4. Solve for ( t ): [ t = frac{20000}{2500} = 8 ]So, the minimum number of years required for the savings to equal the installation cost is 8 years. Part 2: Calculate the net financial benefit or loss after 10 years1. Calculate total savings after 10 years: [ text{Total savings in 10 years} = 2500 times 10 = 25000 ]2. Calculate the increase in the house's value: - The increase in house value due to the geothermal system is 15,000.3. Calculate the total financial gain: [ text{Total gain} = text{Total savings} + text{Increase in house value} = 25000 + 15000 = 40000 ]4. Calculate the net financial benefit or loss: [ text{Net financial benefit/loss} = text{Total gain} - text{Installation cost} = 40000 - 20000 = 20000 ]The homeowner gains 20,000 by installing the geothermal system if they sell the house after 10 years, taking into account the energy savings and the increase in house value.Final Answer: The final answer is 20,000. I hope it is correct.

answer:To solve the given problem, let's address each part step-by-step. Part 1: Determine the year when the number of opioid-related incidents was at its highest.The given equation is:[ I(t) = -2t^2 + 12t + 5 ]This is a quadratic function in the form ( I(t) = at^2 + bt + c ), where ( a = -2 ), ( b = 12 ), and ( c = 5 ).For a quadratic function ( ax^2 + bx + c ), the maximum or minimum value is found at the vertex. Since the coefficient of ( t^2 ) (i.e., ( a )) is negative, the parabola opens downward, and the vertex represents the maximum point.The formula to find the ( t )-coordinate of the vertex is:[ t = -frac{b}{2a} ]Substitute the values of ( a ) and ( b ):[ t = -frac{12}{2(-2)} = -frac{12}{-4} = 3 ]Thus, the maximum number of incidents occurs at ( t = 3 ).To find the maximum number of incidents, substitute ( t = 3 ) back into the equation:[ I(3) = -2(3)^2 + 12(3) + 5 ][ I(3) = -2(9) + 36 + 5 ][ I(3) = -18 + 36 + 5 ][ I(3) = 23 ]Therefore, the maximum number of incidents is 23, occurring in the year ( t = 3 ). Part 2: Find the years when the number of incidents was below 10.We need to solve the inequality:[ I(t) < 10 ]Which means:[ -2t^2 + 12t + 5 < 10 ]Subtract 10 from both sides:[ -2t^2 + 12t + 5 - 10 < 0 ][ -2t^2 + 12t - 5 < 0 ]This can be rewritten as:[ 2t^2 - 12t + 5 > 0 ]To find when this inequality holds, first find the roots of the equation:[ 2t^2 - 12t + 5 = 0 ]Using the quadratic formula ( t = frac{-b pm sqrt{b^2 - 4ac}}{2a} ), with ( a = 2 ), ( b = -12 ), ( c = 5 ):[ t = frac{-(-12) pm sqrt{(-12)^2 - 4 cdot 2 cdot 5}}{2 cdot 2} ][ t = frac{12 pm sqrt{144 - 40}}{4} ][ t = frac{12 pm sqrt{104}}{4} ][ t = frac{12 pm 2sqrt{26}}{4} ][ t = frac{6 pm sqrt{26}}{2} ]The roots are:[ t_1 = frac{6 + sqrt{26}}{2} ][ t_2 = frac{6 - sqrt{26}}{2} ]Since the parabola opens upwards, the inequality ( 2t^2 - 12t + 5 > 0 ) is satisfied outside the interval ([t_1, t_2]).Thus, the solution is:[ t < frac{6 - sqrt{26}}{2} quad text{or} quad t > frac{6 + sqrt{26}}{2} ]Calculating the approximate values:[ sqrt{26} approx 5.1 ]So:[ t_1 approx frac{6 + 5.1}{2} = frac{11.1}{2} approx 5.55 ][ t_2 approx frac{6 - 5.1}{2} = frac{0.9}{2} approx 0.45 ]Interpreting in the context of years, the number of incidents was below 10 before year 0.45 and after year 5.55. Since ( t ) must be an integer, incidents are below 10 in the years 0 and 6 onwards.Final Answer: The final answer is that the maximum number of incidents, 23, occurred in year 3. The number of incidents was below 10 in years 0 and after year 6. I hope it is