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Simplifying 6x2 + 31x + 5 = (6x + -1) Reorder the terms: 5 + 31x + 6x2 = (6x + -1) Reorder the terms: 5 + 31x + 6x2 = (-1 + 6x) Remove parenthesis around (-1 + 6x) 5 + 31x + 6x2 = -1 + 6x Solving 5 + 31x + 6x2 = -1 + 6x Solving for variable 'x'. Reorder the terms: 5 + 1 + 31x + -6x + 6x2 = -1 + 6x + 1 + -6x Combine like terms: 5 + 1 = 6 6 + 31x + -6x + 6x2 = -1 + 6x + 1 + -6x Combine like terms: 31x + -6x = 25x 6 + 25x + 6x2 = -1 + 6x + 1 + -6x Reorder the terms: 6 + 25x + 6x2 = -1 + 1 + 6x + -6x Combine like terms: -1 + 1 = 0 6 + 25x + 6x2 = 0 + 6x + -6x 6 + 25x + 6x2 = 6x + -6x Combine like terms: 6x + -6x = 0 6 + 25x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 1 + 4.166666667x + x2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + 4.166666667x + -1 + x2 = 0 + -1 Reorder the terms: 1 + -1 + 4.166666667x + x2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 4.166666667x + x2 = 0 + -1 4.166666667x + x2 = 0 + -1 Combine like terms: 0 + -1 = -1 4.166666667x + x2 = -1 The x term is 4.166666667x. Take half its coefficient (2.083333334). Square it (4.340277781) and add it to both sides. Add '4.340277781' to each side of the equation. 4.166666667x + 4.340277781 + x2 = -1 + 4.340277781 Reorder the terms: 4.340277781 + 4.166666667x + x2 = -1 + 4.340277781 Combine like terms: -1 + 4.340277781 = 3.340277781 4.340277781 + 4.166666667x + x2 = 3.340277781 Factor a perfect square on the left side: (x + 2.083333334)(x + 2.083333334) = 3.340277781 Calculate the square root of the right side: 1.827642684 Break this problem into two subproblems by setting (x + 2.083333334) equal to 1.827642684 and -1.827642684.Subproblem 1
x + 2.083333334 = 1.827642684 Simplifying x + 2.083333334 = 1.827642684 Reorder the terms: 2.083333334 + x = 1.827642684 Solving 2.083333334 + x = 1.827642684 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.083333334' to each side of the equation. 2.083333334 + -2.083333334 + x = 1.827642684 + -2.083333334 Combine like terms: 2.083333334 + -2.083333334 = 0.000000000 0.000000000 + x = 1.827642684 + -2.083333334 x = 1.827642684 + -2.083333334 Combine like terms: 1.827642684 + -2.083333334 = -0.25569065 x = -0.25569065 Simplifying x = -0.25569065Subproblem 2
x + 2.083333334 = -1.827642684 Simplifying x + 2.083333334 = -1.827642684 Reorder the terms: 2.083333334 + x = -1.827642684 Solving 2.083333334 + x = -1.827642684 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.083333334' to each side of the equation. 2.083333334 + -2.083333334 + x = -1.827642684 + -2.083333334 Combine like terms: 2.083333334 + -2.083333334 = 0.000000000 0.000000000 + x = -1.827642684 + -2.083333334 x = -1.827642684 + -2.083333334 Combine like terms: -1.827642684 + -2.083333334 = -3.910976018 x = -3.910976018 Simplifying x = -3.910976018Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.25569065, -3.910976018}
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