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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Here are the solutions to problems 5, 6, 7, and 8. 5. If and are the roots of the equation 3x^2 - 2x - 1 = 0, find the equation whose roots are ()/() and ()/(). Step 1: Find the sum and product of the roots for the given equation 3x^2 - 2x - 1 = 0. For a quadratic equation ax^2 + bx + c = 0, the sum of roots is + = -(b)/(a) and the product of roots is = (c)/(a). Here, a=3, b=-2, c=-1. Sum of roots: + = -(-2)/(3) = (2)/(3). Product of roots: = (-1)/(3). Step 2: Find the sum of the new roots, S'. The new roots are r_1 = ()/() and r_2 = ()/(). S' = ()/() + ()/() = (^2 + ^2)/() We know that ^2 + ^2 = ( + )^2 - 2. Substitute this into the expression for S': S' = (( + )^2 - 2)/() Substitute the values of ( + ) and : S' = ((2)/(3))^2 - 2(-(1)/(3))-(1)/(3) S' = (4)/(9) + (2)/(3)-(1)/(3) Find a common denominator for the numerator: S' = (4)/(9) + (6)/(9)-(1)/(3) = (10)/(9)-(1)/(3) S' = (10)/(9) × (-(3)/(1)) = -(30)/(9) = -(10)/(3) Step 3: Find the product of the new roots, P'. P' = (()/()) × (()/()) = 1 Step 4: Form the new quadratic equation using the formula x^2 - S'x + P' = 0. x^2 - (-(10)/(3))x + 1 = 0 x^2 + (10)/(3)x + 1 = 0 Multiply the entire equation by 3 to eliminate the fraction: 3x^2 + 10x + 3 = 0 The equation whose roots are ()/() and ()/() 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 of the quadratic equation ax^2 + bx + c = 0. 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). Step 3: Calculate the product of the roots. Product of roots = (c)/(a) = (-2)/(3). The sum of the roots is -(5)/(3) and the product of the roots is -(2)/(3). 7. The product of two consecutive integers is 132. Find the integers. Step 1: Define the integers. Let the first integer be n. The next consecutive integer is n+1. Step 2: Formulate the equation based on the given information. The product of the two consecutive integers is 132: n(n+1) = 132 Step 3: Solve the quadratic equation. n^2 + n = 132 n^2 + n - 132 = 0 Factor the quadratic equation: (n+12)(n-11) = 0 This gives two possible values for n: n = -12 or n = 11 Step 4: Find the pairs of consecutive integers. If n=11, the integers are 11 and 11+1=12. Check: 11 × 12 = 132. If n=-12, the integers are -12 and -12+1=-11. Check: (-12) × (-11) = 132. 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 the dimensions of the rectangle. Let the width of the rectangle be w units. The length exceeds its width by 5 units, so the length l = w+5 units. Step 2: Formulate the equation using the area formula. The area of a rectangle is l × w. Given the area is 104 square units. (w+5)w = 104 Step 3: Solve the quadratic equation. 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 This gives two possible values for w: w = -13 or w = 8 Step 4: Determine the valid dimensions. Since the width of a rectangle cannot be negative, we discard w=-13. So, the width w = 8 units. The length l = w+5 = 8+5 = 13 units. Check: Area = 13 × 8 = 104 square units. The dimensions of the rectangle are length 13 units and width 8 units.