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.
ScanSolve AI Answer
Step 1: Find the sum and product of the roots and for the given equation 2x^2 + 4x + 3 = 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). Given 2x^2 + 4x + 3 = 0, we have a=2, b=4, c=3. Sum of roots: + = -(4)/(2) = -2 Product of roots: = (3)/(2) Step 2: Find the sum of the new roots, which are - 1 and - 1. Let the new roots be r_1 = - 1 and r_2 = - 1. Sum of new roots: r_1 + r_2 = ( - 1) + ( - 1) = + - 2 Substitute the value of + : r_1 + r_2 = -2 - 2 = -4 Step 3: Find the product of the new roots. Product of new roots: r_1r_2 = ( - 1)( - 1) = - - + 1 = - ( + ) + 1 Substitute the values of and + : r_1r_2 = (3)/(2) - (-2) + 1 = (3)/(2) + 2 + 1 = (3)/(2) + 3 = (3)/(2) + (6)/(2) = (9)/(2) Step 4: Form the new quadratic equation. A quadratic equation with roots r_1 and r_2 is given by x^2 - (r_1 + r_2)x + r_1r_2 = 0. Substitute the sum and product of the new roots: x^2 - (-4)x + (9)/(2) = 0 x^2 + 4x + (9)/(2) = 0 To eliminate the fraction, multiply the entire equation by 2: 2(x^2 + 4x + (9)/(2)) = 2(0) 2x^2 + 8x + 9 = 0 Comparing this with the given options, option (b) matches our result. The final answer is 2x^2 + 8x + 9 = 0.