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Simplifying -8x2 + 5x + 31 = 0 Reorder the terms: 31 + 5x + -8x2 = 0 Solving 31 + 5x + -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 + -0.625x + x2 = 0 Move the constant term to the right: Add '3.875' to each side of the equation. -3.875 + -0.625x + 3.875 + x2 = 0 + 3.875 Reorder the terms: -3.875 + 3.875 + -0.625x + x2 = 0 + 3.875 Combine like terms: -3.875 + 3.875 = 0.000 0.000 + -0.625x + x2 = 0 + 3.875 -0.625x + x2 = 0 + 3.875 Combine like terms: 0 + 3.875 = 3.875 -0.625x + x2 = 3.875 The x term is -0.625x. Take half its coefficient (-0.3125). Square it (0.09765625) and add it to both sides. Add '0.09765625' to each side of the equation. -0.625x + 0.09765625 + x2 = 3.875 + 0.09765625 Reorder the terms: 0.09765625 + -0.625x + x2 = 3.875 + 0.09765625 Combine like terms: 3.875 + 0.09765625 = 3.97265625 0.09765625 + -0.625x + x2 = 3.97265625 Factor a perfect square on the left side: (x + -0.3125)(x + -0.3125) = 3.97265625 Calculate the square root of the right side: 1.99315234 Break this problem into two subproblems by setting (x + -0.3125) equal to 1.99315234 and -1.99315234.Subproblem 1
x + -0.3125 = 1.99315234 Simplifying x + -0.3125 = 1.99315234 Reorder the terms: -0.3125 + x = 1.99315234 Solving -0.3125 + x = 1.99315234 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + x = 1.99315234 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + x = 1.99315234 + 0.3125 x = 1.99315234 + 0.3125 Combine like terms: 1.99315234 + 0.3125 = 2.30565234 x = 2.30565234 Simplifying x = 2.30565234Subproblem 2
x + -0.3125 = -1.99315234 Simplifying x + -0.3125 = -1.99315234 Reorder the terms: -0.3125 + x = -1.99315234 Solving -0.3125 + x = -1.99315234 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + x = -1.99315234 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + x = -1.99315234 + 0.3125 x = -1.99315234 + 0.3125 Combine like terms: -1.99315234 + 0.3125 = -1.68065234 x = -1.68065234 Simplifying x = -1.68065234Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.30565234, -1.68065234}
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