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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The given quadratic equation is 3x^2 + 5x - 1 = 0. Let and be the roots of this equation. For a quadratic equation ax^2 + bx + c = 0, the sum of the roots is + = -(b)/(a) and the product of the roots is = (c)/(a). From 3x^2 + 5x - 1 = 0, we have a=3, b=5, c=-1. The sum of the roots is: + = -(5)/(3) The product of the roots is: = (-1)/(3) To construct a new quadratic equation with roots r_1 and r_2, the general form is x^2 - (r_1 + r_2)x + r_1 r_2 = 0. a) Roots are 5, 5 Step 1: Calculate the sum of the new roots. S' = 5 + 5 = 5( + ) = 5(-(5)/(3)) = -(25)/(3) Step 2: Calculate the product of the new roots. P' = (5)(5) = 25 = 25(-(1)/(3)) = -(25)/(3) Step 3: Form the new quadratic equation. x^2 - S'x + P' = 0 x^2 - (-(25)/(3))x + (-(25)/(3)) = 0 x^2 + (25)/(3)x - (25)/(3) = 0 Multiply by 3 to clear the denominators: 3x^2 + 25x - 25 = 0 b) Roots are ^2, ^2 Step 1: Calculate the sum of the new roots. S' = ^2 + ^2 = ( + )^2 - 2 S' = (-(5)/(3))^2 - 2(-(1)/(3)) = (25)/(9) + (2)/(3) = (25)/(9) + (6)/(9) = (31)/(9) Step 2: Calculate the product of the new roots. P' = ^2 ^2 = ()^2 = (-(1)/(3))^2 = (1)/(9) Step 3: Form the new quadratic equation. x^2 - S'x + P' = 0 x^2 - (31)/(9)x + (1)/(9) = 0 Multiply by 9 to clear the denominators: 9x^2 - 31x + 1 = 0 c) Roots are (1)/(), (1)/() Step 1: Calculate the sum of the new roots. S' = (1)/() + (1)/() = ( + )/() S' = (-5)/(3)-(1)/(3) = 5 Step 2: Calculate the product of the new roots. P' = ((1)/())((1)/()) = (1)/() P' = (1)/(-1)3 = -3 Step 3: Form the new quadratic equation. x^2 - S'x + P' = 0 x^2 - 5x - 3 = 0 x^2 - 5x - 3 = 0 d) Roots are + (1)/(), + (1)/() Step 1: Calculate the sum of the new roots. S' = ( + (1)/()) + ( + (1)/()) = ( + ) + ((1)/() + (1)/()) We know + = -(5)/(3) and (1)/() + (1)/() = ( + )/() = (-5)/(3)-(1)/(3) = 5. S' = -(5)/(3) + 5 = -(5)/(3) + (15)/(3) = (10)/(3) Step 2: Calculate the product of the new roots. P' = ( + (1)/())( + (1)/()) = + ((1)/()) + ((1)/()) + ((1)/())((1)/()) P' = + 1 + 1 + (1)/() = + 2 + (1)/() P' = (-(1)/(3)) + 2 + (1)/(-1)3 = -(1)/(3) + 2 - 3 P' = -(1)/(3) - 1 = -(1)/(3) - (3)/(3) = -(4)/(3) Step 3: Form the new quadratic equation. x^2 - S'x + P' = 0 x^2 - (10)/(3)x - (4)/(3) = 0 Multiply by 3 to clear the denominators: 3x^2 - 10x - 4 = 0 Got more? Send 'em!