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question:A mentor and trainer noticed a young player's talent in a special game that involves solving complex mathematical puzzles. To help the player develop, the mentor devised a unique training exercise that involves algebraic expressions and equations. The exercise is designed to improve both the player's problem-solving skills and their ability to think strategically.1. The mentor set up a polynomial equation as part of the training: (2x^3 - 3x^2 - 11x + 6 = 0). The player needs to find all the real roots of the equation to progress to the next stage of training. 2. Once the player finds the real roots of the polynomial, the mentor challenges them with a rational expression problem. Simplify the following expression and state any restrictions on the variable: (frac{x^2 - 9}{x^2 - x - 6}).The player's ability to solve these problems will determine their readiness for the advanced stages of training.
answer:To tackle the mentor's training exercise, we will solve each part of the problem step-by-step. Part 1: Finding the Real Roots of the Polynomial EquationThe polynomial equation given is:[ 2x^3 - 3x^2 - 11x + 6 = 0 ]First, we will try to find the rational roots using the Rational Root Theorem. According to this theorem, any rational root, ( frac{p}{q} ), of the polynomial must be such that ( p ) divides the constant term ((6)) and ( q ) divides the leading coefficient ((2)).Possible values for ( p ) (factors of 6) are: ( pm 1, pm 2, pm 3, pm 6 ).Possible values for ( q ) (factors of 2) are: ( pm 1, pm 2 ).Thus, possible rational roots are: [ pm 1, pm 2, pm 3, pm 6, pm frac{1}{2}, pm frac{3}{2} ]We will test these values by substituting them into the polynomial to see if they yield zero.Testing ( x = 1 ):[ 2(1)^3 - 3(1)^2 - 11(1) + 6 = 2 - 3 - 11 + 6 = -6 neq 0 ]Testing ( x = -1 ):[ 2(-1)^3 - 3(-1)^2 - 11(-1) + 6 = -2 - 3 + 11 + 6 = 12 neq 0 ]Testing ( x = 2 ):[ 2(2)^3 - 3(2)^2 - 11(2) + 6 = 16 - 12 - 22 + 6 = -12 neq 0 ]Testing ( x = -2 ):[ 2(-2)^3 - 3(-2)^2 - 11(-2) + 6 = -16 - 12 + 22 + 6 = 0 ]We have found a root: ( x = -2 ).Next, we perform synthetic division of the polynomial by ( x + 2 ) to factor the polynomial.Using synthetic division with root ( x = -2 ):```-2 | 2 -3 -11 6 | -4 14 -6 --------------------- 2 -7 3 0```The quotient is ( 2x^2 - 7x + 3 ).Now, solve ( 2x^2 - 7x + 3 = 0 ) using the quadratic formula:[ x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ]where ( a = 2 ), ( b = -7 ), ( c = 3 ).[ x = frac{-(-7) pm sqrt{(-7)^2 - 4 cdot 2 cdot 3}}{2 cdot 2} ][ x = frac{7 pm sqrt{49 - 24}}{4} ][ x = frac{7 pm sqrt{25}}{4} ][ x = frac{7 pm 5}{4} ]The solutions are:[ x = frac{7 + 5}{4} = 3 ][ x = frac{7 - 5}{4} = frac{1}{2} ]Thus, the real roots of the polynomial are ( x = -2, x = 3, x = frac{1}{2} ). Part 2: Simplifying the Rational ExpressionThe given rational expression is:[ frac{x^2 - 9}{x^2 - x - 6} ]First, factor both the numerator and the denominator.Numerator: ( x^2 - 9 ) is a difference of squares.[ x^2 - 9 = (x - 3)(x + 3) ]Denominator: ( x^2 - x - 6 ) factors as:[ x^2 - x - 6 = (x - 3)(x + 2) ]The rational expression becomes:[ frac{(x - 3)(x + 3)}{(x - 3)(x + 2)} ]Cancel the common factor ( (x - 3) ), but note that ( x neq 3 )
question:As a paralegal responsible for evaluating legal compliance, you are tasked with analyzing the number of legal documents required in two different countries, Country A and Country B, to ensure compliance in a specific field of law. The number of legal documents, ( D_A ) and ( D_B ), required in Country A and Country B, respectively, can be modeled by the following quadratic equations:1. ( D_A = x^2 - 4x + 6 )2. ( D_B = 2x^2 - 8x + 8 )where ( x ) represents the complexity level of the legal compliance issue, ranging from 1 to 5.(a) Determine the complexity level ( x ) at which the number of documents required in Country A and Country B are equal.(b) Once you find the complexity level from part (a), calculate the number of documents required in each country at that complexity level.
answer:To solve the problem, we need to find the complexity level ( x ) at which the number of legal documents required in Country A, ( D_A ), is equal to the number of legal documents required in Country B, ( D_B ). This involves solving the equation ( D_A = D_B ).The given equations are:1. ( D_A = x^2 - 4x + 6 )2. ( D_B = 2x^2 - 8x + 8 ) (a) Find the complexity level ( x ) such that ( D_A = D_B ).Set the two equations equal to each other:[ x^2 - 4x + 6 = 2x^2 - 8x + 8 ]Rearrange the equation to form a quadratic equation:[ 0 = 2x^2 - 8x + 8 - x^2 + 4x - 6 ]Combine like terms:[ 0 = x^2 - 4x + 2 ]This simplifies to:[ x^2 - 4x + 2 = 0 ]Now, solve this quadratic equation using the quadratic formula ( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ), where ( a = 1 ), ( b = -4 ), and ( c = 2 ).Calculate the discriminant:[ b^2 - 4ac = (-4)^2 - 4 cdot 1 cdot 2 = 16 - 8 = 8 ]Since the discriminant is positive, there are two real solutions. Calculate these solutions:[ x = frac{-(-4) pm sqrt{8}}{2 cdot 1} ][ x = frac{4 pm sqrt{8}}{2} ][ x = frac{4 pm 2sqrt{2}}{2} ][ x = 2 pm sqrt{2} ]Since ( x ) must be in the range from 1 to 5 (as given in the problem), we need to check which of these solutions are valid:- ( x = 2 + sqrt{2} ) is approximately ( 2 + 1.414 approx 3.414 ), which is within the range.- ( x = 2 - sqrt{2} ) is approximately ( 2 - 1.414 approx 0.586 ), which is outside the range.Hence, the valid solution is ( x = 2 + sqrt{2} ). (b) Calculate the number of documents required at this complexity level ( x = 2 + sqrt{2} ).Substitute ( x = 2 + sqrt{2} ) into either ( D_A ) or ( D_B ).First, calculate for ( D_A ):[ D_A = (2 + sqrt{2})^2 - 4(2 + sqrt{2}) + 6 ]Calculate each term separately:1. ( (2 + sqrt{2})^2 = 4 + 4sqrt{2} + 2 = 6 + 4sqrt{2} )2. ( 4(2 + sqrt{2}) = 8 + 4sqrt{2} )Now, substitute back into the equation:[ D_A = (6 + 4sqrt{2}) - (8 + 4sqrt{2}) + 6 ][ D_A = 6 + 4sqrt{2} - 8 - 4sqrt{2} + 6 ][ D_A = 4 ]Now, calculate for ( D_B ):[ D_B = 2(2 + sqrt{2})^2 - 8(2 + sqrt{2}) + 8 ]Calculate each term separately:1. ( 2(6 + 4sqrt{2}) = 12 + 8sqrt{2} )2. ( 8(2 + sqrt{2}) = 16 + 8sqrt{2} )Now, substitute back into the equation:[ D_B = 12 + 8sqrt{2} - (16 + 8sqrt{2}) + 8 ][ D_B = 12 + 8sqrt{2} - 16 - 8sqrt{2} + 8 ][ D_B = 4 ]Therefore, at the complexity level ( x = 2 + sqrt{2} ), the number of documents required in both Country A and Country B is 4.Final
question:A Brazilian historian is analyzing the trade agreements between Brazil and Japan over the years. She finds a polynomial equation that models the annual trade growth (in billions of dollars) between the two countries over a certain period. The polynomial is given by ( P(x) = x^3 - 4x^2 + 7x - 6 ), where ( x ) represents the number of years since the beginning of the analysis.1. Solve the polynomial equation ( P(x) = 0 ) to determine the years when the trade growth was exactly zero.2. Additionally, the historian is interested in determining the average annual trade growth rate. If the polynomial ( P(x) ) represents the accumulated trade growth over ( x ) years, find the expression for the average annual growth rate over ( x ) years, and simplify it as much as possible.
answer:To solve the given problem, we proceed step-by-step. Step 1: Solve the polynomial equation ( P(x) = 0 ).The polynomial given is ( P(x) = x^3 - 4x^2 + 7x - 6 ). We need to find the roots of this equation, which are the values of ( x ) for which ( P(x) = 0 ).Finding the Roots:One approach to finding the roots of a cubic polynomial is to use the Rational Root Theorem, which suggests that any rational root, in the form ( frac{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).The possible rational roots are ( pm 1, pm 2, pm 3, pm 6 ).We test these values in the polynomial.- Testing ( x = 1 ): [ P(1) = 1^3 - 4 times 1^2 + 7 times 1 - 6 = 1 - 4 + 7 - 6 = -2 ] ( x = 1 ) is not a root.- Testing ( x = 2 ): [ P(2) = 2^3 - 4 times 2^2 + 7 times 2 - 6 = 8 - 16 + 14 - 6 = 0 ] ( x = 2 ) is a root.Now that we have found one root, ( x = 2 ), we can use polynomial division to factor ( P(x) ) by ( (x - 2) ).Polynomial Division:Divide ( P(x) = x^3 - 4x^2 + 7x - 6 ) by ( (x - 2) ):1. Divide the leading term ( x^3 ) by ( x ) to get ( x^2 ).2. Multiply ( x^2 ) by ( (x - 2) ) to get ( x^3 - 2x^2 ).3. Subtract ( (x^3 - 2x^2) ) from ( P(x) ), leaving ( -2x^2 + 7x ).4. Divide (-2x^2) by ( x ) to get (-2x).5. Multiply (-2x) by ( (x - 2) ) to get (-2x^2 + 4x).6. Subtract (-2x^2 + 4x) from (-2x^2 + 7x), leaving ( 3x ).7. Divide (3x) by ( x ) to get (3).8. Multiply (3) by ( (x - 2) ) to get (3x - 6).9. Subtract (3x - 6) from (3x - 6), leaving a remainder of 0.The quotient is ( x^2 - 2x + 3 ).Thus, ( P(x) = (x - 2)(x^2 - 2x + 3) ).Finding the Roots of the Quadratic:The quadratic equation is ( x^2 - 2x + 3 = 0 ).Use the quadratic formula:[x = frac{-b pm sqrt{b^2 - 4ac}}{2a}]Where ( a = 1 ), ( b = -2 ), and ( c = 3 ).[x = frac{-(-2) pm sqrt{(-2)^2 - 4 cdot 1 cdot 3}}{2 cdot 1}][x = frac{2 pm sqrt{4 - 12}}{2}][x = frac{2 pm sqrt{-8}}{2}][x = frac{2 pm 2isqrt{2}}{2}][x = 1 pm isqrt{2}]The roots of the equation are ( x = 2 ), ( x = 1 + isqrt{2} ), and ( x = 1 - isqrt{2} ). Step 2: Determine the average annual trade growth rate.The polynomial ( P(x) = x^3 - 4x^2 + 7x - 6 )
question:A Costa Rican citizen is excited about women's international football and decides to create a small exhibition about the growth of women's football teams worldwide since 2004, using quadratic equations to model the potential growth of teams over time. 1. Suppose in 2004, the number of women's football teams worldwide was estimated to be 3,600. The citizen estimates that the number of teams has been increasing according to the quadratic model ( T(t) = at^2 + bt + 3600 ), where ( t ) is the number of years since 2004. If in 2014 there were approximately 4,200 teams, find the values of ( a ) and ( b ) given an additional data point that in 2009, there were 4,000 teams.2. Using the model ( T(t) = at^2 + bt + 3600 ) with the values of ( a ) and ( b ) found in the previous part, determine the number of teams predicted for the year 2024.
answer:To solve the problem, we need to find the values of ( a ) and ( b ) in the quadratic equation ( T(t) = at^2 + bt + 3600 ) using the given data points. Step 1: Set Up Equations from Given Data PointsWe have two data points:- In 2009 (( t = 5 )), ( T(5) = 4000 ).- In 2014 (( t = 10 )), ( T(10) = 4200 ).Substitute these into the quadratic equation:1. For ( t = 5 ), the equation is: [ a(5)^2 + b(5) + 3600 = 4000 ] Simplifying, we get: [ 25a + 5b + 3600 = 4000 ] [ 25a + 5b = 400 ] [ 5a + b = 80 quad text{(Equation 1)} ]2. For ( t = 10 ), the equation is: [ a(10)^2 + b(10) + 3600 = 4200 ] Simplifying, we get: [ 100a + 10b + 3600 = 4200 ] [ 100a + 10b = 600 ] [ 10a + b = 60 quad text{(Equation 2)} ] Step 2: Solve the System of EquationsNow, solve the system formed by Equation 1 and Equation 2:Equation 1:[ 5a + b = 80 ]Equation 2:[ 10a + b = 60 ]Subtract Equation 1 from Equation 2:[(10a + b) - (5a + b) = 60 - 80][5a = -20][a = -4]Substitute ( a = -4 ) back into Equation 1:[5(-4) + b = 80][-20 + b = 80][b = 100]Thus, the values are ( a = -4 ) and ( b = 100 ). Step 3: Predict the Number of Teams in 2024Now, use the quadratic model with ( a = -4 ) and ( b = 100 ) to predict the number of teams in 2024 (( t = 20 )):[T(20) = -4(20)^2 + 100(20) + 3600][= -4(400) + 2000 + 3600][= -1600 + 2000 + 3600][= 4000]Final Answer: The final answer is 4000. I hope it is correct.
question:As a small business owner looking to improve customer experience, you've decided to revise the product manuals for your best-selling product. You plan to print a certain number of manuals, and the cost of printing each manual depends on the number of pages. The printing company charges a base fee of 50 and an additional 0.20 per page for the first 100 pages. For any pages beyond 100, the cost is reduced to 0.15 per page. 1. If you want the total cost to be under 200, write and solve an inequality to find the maximum number of pages you can include in a single manual.2. After determining the maximum number of pages, you realize that you need to include additional content that will increase the number of pages by 20%. Write a polynomial equation to represent the new total number of pages, and solve it to find the exact number of pages in the revised manual.
answer:Let's solve the problem step-by-step. Problem 1: Finding the Maximum Number of PagesThe total cost consists of a base fee plus per-page charges. For a manual with ( x ) pages, the cost structure is as follows:- Base fee: 50- Cost for the first 100 pages: ( 0.20 times x ) (for ( x leq 100 )) - Cost for pages beyond 100: ( 0.20 times 100 + 0.15 times (x - 100) ) (for ( x > 100 ))We want the total cost to be under 200:[ text{Base fee} + text{Page cost} < 200 ]# Case 1: ( x leq 100 )[ 50 + 0.20x < 200 ]Subtract 50 from both sides:[ 0.20x < 150 ]Divide by 0.20:[ x < 750 ]Since ( x leq 100 ), this case allows up to 100 pages.# Case 2: ( x > 100 )The cost becomes:[ 50 + 0.20 times 100 + 0.15 times (x - 100) < 200 ]Simplify the equation:[ 50 + 20 + 0.15(x - 100) < 200 ][ 70 + 0.15x - 15 < 200 ][ 0.15x + 55 < 200 ]Subtract 55 from both sides:[ 0.15x < 145 ]Divide by 0.15:[ x < frac{145}{0.15} ][ x < 966.67 ]Since ( x ) must be an integer, ( x leq 966 ).Combining both cases, the maximum number of pages is 100 when ( x leq 100 ). However, since ( x > 100 ) is possible up to 966, the maximum allowable number of pages is 966. Problem 2: Adding 20% More PagesLet ( y ) be the new total number of pages. The revised manual has pages increased by 20%:[ y = x + 0.2x ][ y = 1.2x ]Using ( x = 966 ):[ y = 1.2 times 966 ][ y = 1159.2 ]Since the number of pages must be an integer, we round to the nearest whole number:[ y = 1159 ] Final AnswerThe final answer is ( 1159 ). I hope it is correct.
question:As a rural development policymaker in Norway, you are tasked with optimizing the allocation of resources for sustainable farming communities. You plan to distribute funds across two main initiatives: renewable energy (E) and sustainable agriculture (A). The total fund available for distribution is 1,000,000 NOK.1. The relationship between the funds allocated to renewable energy and sustainable agriculture can be modeled by the quadratic equation ( E^2 + 2A = 500,000 ). Additionally, the sum of the funds allocated to both initiatives must equal the total fund available: ( E + A = 1,000,000 ). Find the amount of funds allocated to each initiative.2. The Norwegian government has set a condition that the funds allocated to renewable energy should not be less than 40% of the total fund. Based on the solution of the quadratic equation, determine the inequality that represents this condition and verify whether the solution satisfies this requirement.
answer:To solve the given problem, we need to find the values of ( E ) and ( A ) that satisfy both equations:1. ( E^2 + 2A = 500,000 )2. ( E + A = 1,000,000 )Step 1: Solve the system of equations.From equation (2), express ( A ) in terms of ( E ):[ A = 1,000,000 - E ]Substitute this expression for ( A ) into equation (1):[ E^2 + 2(1,000,000 - E) = 500,000 ]Simplify the equation:[ E^2 + 2,000,000 - 2E = 500,000 ]Rearrange terms to form a quadratic equation:[ E^2 - 2E + 1,500,000 = 0 ]Step 2: Solve the quadratic equation for ( E ).This is a standard quadratic equation of the form ( ax^2 + bx + c = 0 ), where ( a = 1 ), ( b = -2 ), and ( c = 1,500,000 ).The quadratic formula is given by:[ E = frac{-b pm sqrt{b^2 - 4ac}}{2a} ]Substitute the values for ( a ), ( b ), and ( c ):[ E = frac{-(-2) pm sqrt{(-2)^2 - 4 cdot 1 cdot 1,500,000}}{2 cdot 1} ][ E = frac{2 pm sqrt{4 - 6,000,000}}{2} ][ E = frac{2 pm sqrt{-5,999,996}}{2} ]Since the discriminant ((-5,999,996)) is negative, there are no real solutions for ( E ) and ( A ) under the given equations, indicating an error in formulating the problem or conditions.Step 3: Check the condition for renewable energy allocation.The condition states that funds for renewable energy should not be less than 40% of the total fund:[ E geq 0.4 times 1,000,000 = 400,000 ]However, since there are no real solutions for ( E ) from our calculations, we cannot verify this condition with the given setup.Final Answer: The final answer is that there are no real solutions for ( E ) and ( A ) due to a negative discriminant in the quadratic equation. Please recheck the problem constraints or equations. I hope it is correct.