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Simplifying 16x2 + -31x + 5 = 0 Reorder the terms: 5 + -31x + 16x2 = 0 Solving 5 + -31x + 16x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.3125 + -1.9375x + x2 = 0 Move the constant term to the right: Add '-0.3125' to each side of the equation. 0.3125 + -1.9375x + -0.3125 + x2 = 0 + -0.3125 Reorder the terms: 0.3125 + -0.3125 + -1.9375x + x2 = 0 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + -1.9375x + x2 = 0 + -0.3125 -1.9375x + x2 = 0 + -0.3125 Combine like terms: 0 + -0.3125 = -0.3125 -1.9375x + x2 = -0.3125 The x term is -1.9375x. Take half its coefficient (-0.96875). Square it (0.9384765625) and add it to both sides. Add '0.9384765625' to each side of the equation. -1.9375x + 0.9384765625 + x2 = -0.3125 + 0.9384765625 Reorder the terms: 0.9384765625 + -1.9375x + x2 = -0.3125 + 0.9384765625 Combine like terms: -0.3125 + 0.9384765625 = 0.6259765625 0.9384765625 + -1.9375x + x2 = 0.6259765625 Factor a perfect square on the left side: (x + -0.96875)(x + -0.96875) = 0.6259765625 Calculate the square root of the right side: 0.791186806 Break this problem into two subproblems by setting (x + -0.96875) equal to 0.791186806 and -0.791186806.Subproblem 1
x + -0.96875 = 0.791186806 Simplifying x + -0.96875 = 0.791186806 Reorder the terms: -0.96875 + x = 0.791186806 Solving -0.96875 + x = 0.791186806 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.96875' to each side of the equation. -0.96875 + 0.96875 + x = 0.791186806 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + x = 0.791186806 + 0.96875 x = 0.791186806 + 0.96875 Combine like terms: 0.791186806 + 0.96875 = 1.759936806 x = 1.759936806 Simplifying x = 1.759936806Subproblem 2
x + -0.96875 = -0.791186806 Simplifying x + -0.96875 = -0.791186806 Reorder the terms: -0.96875 + x = -0.791186806 Solving -0.96875 + x = -0.791186806 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.96875' to each side of the equation. -0.96875 + 0.96875 + x = -0.791186806 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + x = -0.791186806 + 0.96875 x = -0.791186806 + 0.96875 Combine like terms: -0.791186806 + 0.96875 = 0.177563194 x = 0.177563194 Simplifying x = 0.177563194Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.759936806, 0.177563194}
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