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Simplifying 32x2 + 64x + 23 = 0 Reorder the terms: 23 + 64x + 32x2 = 0 Solving 23 + 64x + 32x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. 0.71875 + 2x + x2 = 0 Move the constant term to the right: Add '-0.71875' to each side of the equation. 0.71875 + 2x + -0.71875 + x2 = 0 + -0.71875 Reorder the terms: 0.71875 + -0.71875 + 2x + x2 = 0 + -0.71875 Combine like terms: 0.71875 + -0.71875 = 0.00000 0.00000 + 2x + x2 = 0 + -0.71875 2x + x2 = 0 + -0.71875 Combine like terms: 0 + -0.71875 = -0.71875 2x + x2 = -0.71875 The x term is 2x. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2x + 1 + x2 = -0.71875 + 1 Reorder the terms: 1 + 2x + x2 = -0.71875 + 1 Combine like terms: -0.71875 + 1 = 0.28125 1 + 2x + x2 = 0.28125 Factor a perfect square on the left side: (x + 1)(x + 1) = 0.28125 Calculate the square root of the right side: 0.530330086 Break this problem into two subproblems by setting (x + 1) equal to 0.530330086 and -0.530330086.Subproblem 1
x + 1 = 0.530330086 Simplifying x + 1 = 0.530330086 Reorder the terms: 1 + x = 0.530330086 Solving 1 + x = 0.530330086 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 0.530330086 + -1 Combine like terms: 1 + -1 = 0 0 + x = 0.530330086 + -1 x = 0.530330086 + -1 Combine like terms: 0.530330086 + -1 = -0.469669914 x = -0.469669914 Simplifying x = -0.469669914Subproblem 2
x + 1 = -0.530330086 Simplifying x + 1 = -0.530330086 Reorder the terms: 1 + x = -0.530330086 Solving 1 + x = -0.530330086 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -0.530330086 + -1 Combine like terms: 1 + -1 = 0 0 + x = -0.530330086 + -1 x = -0.530330086 + -1 Combine like terms: -0.530330086 + -1 = -1.530330086 x = -1.530330086 Simplifying x = -1.530330086Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.469669914, -1.530330086}
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