This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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5. If and are the roots of the equation 3x^2 - 2x - 1 = 0, find the equation whose roots are ()/() and ()/(). Step 1: Identify the coefficients and find the sum and product of the original roots. For 3x^2 - 2x - 1 = 0, we have a = 3, b = -2, c = -1. Sum of roots: + = -(b)/(a) = -((-2))/(3) = (2)/(3). Product of roots: = (c)/(a) = (-1)/(3). Step 2: Find the sum of the new roots. Let the new roots be r_1 = ()/() and r_2 = ()/(). Sum of new roots: S' = ()/() + ()/() = (^2 + ^2)/(). We know that ^2 + ^2 = ( + )^2 - 2. S' = (( + )^2 - 2)/() Substitute the values of + and : S' = ((2)/(3))^2 - 2(-(1)/(3))-(1)/(3) S' = (4)/(9) + (2)/(3)-(1)/(3) S' = (4)/(9) + (6)/(9)-(1)/(3) S' = (10)/(9)-(1)/(3) S' = (10)/(9) × (-(3)/(1)) S' = -(30)/(9) = -(10)/(3) Step 3: Find the product of the new roots. Product of new roots: P' = (()/()) (()/()) = 1. Step 4: Form the new quadratic equation. The equation is x^2 - S'x + P' = 0. x^2 - (-(10)/(3))x + 1 = 0 x^2 + (10)/(3)x + 1 = 0 Multiply by 3 to clear the fraction: 3x^2 + 10x + 3 = 0 The equation is 3x^2 + 10x + 3 = 0. 6. Without solving the equation, find the sum and product of the roots of 3x^2 + 5x - 2 = 0. Step 1: Identify the coefficients. For 3x^2 + 5x - 2 = 0, we have a = 3, b = 5, c = -2. Step 2: Calculate the sum of the roots. Sum of roots = -(b)/(a) = -(5)/(3). The sum of the roots is -(5)/(3). Step 3: Calculate the product of the roots. Product of roots = (c)/(a) = (-2)/(3). The product of the roots is -(2)/(3). 7. The product of two consecutive integers is 132. Find the integers. Step 1: Set up the equation. Let the first integer be n. The next consecutive integer is n+1. Their product is n(n+1) = 132. Step 2: Solve the quadratic equation. n^2 + n = 132 n^2 + n - 132 = 0 Factor the quadratic equation: We need two numbers that multiply to -132 and add to 1. These numbers are 12 and -11. (n + 12)(n - 11) = 0 So, n = -12 or n = 11. Step 3: Find the pairs of consecutive integers. If n = 11, the integers are 11 and 11+1 = 12. If n = -12, the integers are -12 and -12+1 = -11. The integers are 11 and 12 or -12 and -11. 8. The length of a rectangle exceeds its width by 5 units. If the area is 104 square units, find the dimensions. Step 1: Define variables and set up the equations. Let the width of the rectangle be w units. The length l exceeds the width by 5 units, so l = w + 5. The area A is 104 square units. The formula for the area of a rectangle is A = l × w. (w+5)w = 104 Step 2: Solve the quadratic equation for w. w^2 + 5w = 104 w^2 + 5w - 104 = 0 Factor the quadratic equation: We need two numbers that multiply to -104 and add to 5. These numbers are 13 and -8. (w + 13)(w - 8) = 0 So, w = -13 or w = 8. Step 3: Determine the dimensions. Since width cannot be negative, we take w = 8 units. Length l = w + 5 = 8 + 5 = 13 units. The dimensions are width = 8 units, length = 13 units. 9. Show that if a < 0, y = ax^2 + bx + c has a maximum value when x = -(b)/(2a). Step 1: Rewrite the quadratic equation by completing the square. y = ax^2 + bx + c Factor out a from the terms involving x: y = a(x^2 + (b)/(a)x) + c To complete the square inside the parenthesis, add and subtract ((b)/(2a))^2: y = a(x^2 + (b)/(a)x + ((b)/(2a))^2 - ((b)/(2a))^2) + c y = a((x + (b)/(2a))^2 - (b^2)/(4a^2)) + c Distribute a: y = a(x + (b)/(2a))^2 - a(b^2)/(4a^2) + c y = a(x + (b)/(2a))^2 - (b^2)/(4a) + c y = a(x + (b)/(2a))^2 + (4ac - b^2)/(4a) Step 2: Analyze the expression to find the maximum value. The term a(x + (b)/(2a))^2 is the only term that depends on x. Given that a < 0, the term a(x + (b)/(2a))^2 will always be less than or equal to zero, because (x + (b)/(2a))^2 is always non-negative. For y to have a maximum value, the term a(x + (b)/(2a))^2 must be at its maximum possible value, which is 0. This occurs when (x + (b)/(2a))^2 = 0. x + (b)/(2a) = 0 x = -(b)/(2a) When x = -(b)/(2a), the value of y is y = a(0)^2 + (4ac - b^2)/(4a) = (4ac - b^2)/(4a). Since a < 0, the parabola opens downwards, and the vertex represents the maximum point. Therefore, the maximum value of y occurs at x = -(b)/(2a).