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Simplifying 15x2 + 31x + 12 = 0 Reorder the terms: 12 + 31x + 15x2 = 0 Solving 12 + 31x + 15x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. 0.8 + 2.066666667x + x2 = 0 Move the constant term to the right: Add '-0.8' to each side of the equation. 0.8 + 2.066666667x + -0.8 + x2 = 0 + -0.8 Reorder the terms: 0.8 + -0.8 + 2.066666667x + x2 = 0 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + 2.066666667x + x2 = 0 + -0.8 2.066666667x + x2 = 0 + -0.8 Combine like terms: 0 + -0.8 = -0.8 2.066666667x + x2 = -0.8 The x term is 2.066666667x. Take half its coefficient (1.033333334). Square it (1.067777779) and add it to both sides. Add '1.067777779' to each side of the equation. 2.066666667x + 1.067777779 + x2 = -0.8 + 1.067777779 Reorder the terms: 1.067777779 + 2.066666667x + x2 = -0.8 + 1.067777779 Combine like terms: -0.8 + 1.067777779 = 0.267777779 1.067777779 + 2.066666667x + x2 = 0.267777779 Factor a perfect square on the left side: (x + 1.033333334)(x + 1.033333334) = 0.267777779 Calculate the square root of the right side: 0.517472491 Break this problem into two subproblems by setting (x + 1.033333334) equal to 0.517472491 and -0.517472491.Subproblem 1
x + 1.033333334 = 0.517472491 Simplifying x + 1.033333334 = 0.517472491 Reorder the terms: 1.033333334 + x = 0.517472491 Solving 1.033333334 + x = 0.517472491 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.033333334' to each side of the equation. 1.033333334 + -1.033333334 + x = 0.517472491 + -1.033333334 Combine like terms: 1.033333334 + -1.033333334 = 0.000000000 0.000000000 + x = 0.517472491 + -1.033333334 x = 0.517472491 + -1.033333334 Combine like terms: 0.517472491 + -1.033333334 = -0.515860843 x = -0.515860843 Simplifying x = -0.515860843Subproblem 2
x + 1.033333334 = -0.517472491 Simplifying x + 1.033333334 = -0.517472491 Reorder the terms: 1.033333334 + x = -0.517472491 Solving 1.033333334 + x = -0.517472491 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.033333334' to each side of the equation. 1.033333334 + -1.033333334 + x = -0.517472491 + -1.033333334 Combine like terms: 1.033333334 + -1.033333334 = 0.000000000 0.000000000 + x = -0.517472491 + -1.033333334 x = -0.517472491 + -1.033333334 Combine like terms: -0.517472491 + -1.033333334 = -1.550805825 x = -1.550805825 Simplifying x = -1.550805825Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.515860843, -1.550805825}
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