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Simplifying 8x2 + 68x + 31 = 0 Reorder the terms: 31 + 68x + 8x2 = 0 Solving 31 + 68x + 8x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 3.875 + 8.5x + x2 = 0 Move the constant term to the right: Add '-3.875' to each side of the equation. 3.875 + 8.5x + -3.875 + x2 = 0 + -3.875 Reorder the terms: 3.875 + -3.875 + 8.5x + x2 = 0 + -3.875 Combine like terms: 3.875 + -3.875 = 0.000 0.000 + 8.5x + x2 = 0 + -3.875 8.5x + x2 = 0 + -3.875 Combine like terms: 0 + -3.875 = -3.875 8.5x + x2 = -3.875 The x term is 8.5x. Take half its coefficient (4.25). Square it (18.0625) and add it to both sides. Add '18.0625' to each side of the equation. 8.5x + 18.0625 + x2 = -3.875 + 18.0625 Reorder the terms: 18.0625 + 8.5x + x2 = -3.875 + 18.0625 Combine like terms: -3.875 + 18.0625 = 14.1875 18.0625 + 8.5x + x2 = 14.1875 Factor a perfect square on the left side: (x + 4.25)(x + 4.25) = 14.1875 Calculate the square root of the right side: 3.766629793 Break this problem into two subproblems by setting (x + 4.25) equal to 3.766629793 and -3.766629793.Subproblem 1
x + 4.25 = 3.766629793 Simplifying x + 4.25 = 3.766629793 Reorder the terms: 4.25 + x = 3.766629793 Solving 4.25 + x = 3.766629793 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + x = 3.766629793 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + x = 3.766629793 + -4.25 x = 3.766629793 + -4.25 Combine like terms: 3.766629793 + -4.25 = -0.483370207 x = -0.483370207 Simplifying x = -0.483370207Subproblem 2
x + 4.25 = -3.766629793 Simplifying x + 4.25 = -3.766629793 Reorder the terms: 4.25 + x = -3.766629793 Solving 4.25 + x = -3.766629793 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + x = -3.766629793 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + x = -3.766629793 + -4.25 x = -3.766629793 + -4.25 Combine like terms: -3.766629793 + -4.25 = -8.016629793 x = -8.016629793 Simplifying x = -8.016629793Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.483370207, -8.016629793}
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