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Simplifying -16x2 + 32x + 5 = 6 Reorder the terms: 5 + 32x + -16x2 = 6 Solving 5 + 32x + -16x2 = 6 Solving for variable 'x'. Reorder the terms: 5 + -6 + 32x + -16x2 = 6 + -6 Combine like terms: 5 + -6 = -1 -1 + 32x + -16x2 = 6 + -6 Combine like terms: 6 + -6 = 0 -1 + 32x + -16x2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 0.0625 + -2x + x2 = 0 Move the constant term to the right: Add '-0.0625' to each side of the equation. 0.0625 + -2x + -0.0625 + x2 = 0 + -0.0625 Reorder the terms: 0.0625 + -0.0625 + -2x + x2 = 0 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + -2x + x2 = 0 + -0.0625 -2x + x2 = 0 + -0.0625 Combine like terms: 0 + -0.0625 = -0.0625 -2x + x2 = -0.0625 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 = -0.0625 + 1 Reorder the terms: 1 + -2x + x2 = -0.0625 + 1 Combine like terms: -0.0625 + 1 = 0.9375 1 + -2x + x2 = 0.9375 Factor a perfect square on the left side: (x + -1)(x + -1) = 0.9375 Calculate the square root of the right side: 0.968245837 Break this problem into two subproblems by setting (x + -1) equal to 0.968245837 and -0.968245837.Subproblem 1
x + -1 = 0.968245837 Simplifying x + -1 = 0.968245837 Reorder the terms: -1 + x = 0.968245837 Solving -1 + x = 0.968245837 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 = 0.968245837 + 1 Combine like terms: -1 + 1 = 0 0 + x = 0.968245837 + 1 x = 0.968245837 + 1 Combine like terms: 0.968245837 + 1 = 1.968245837 x = 1.968245837 Simplifying x = 1.968245837Subproblem 2
x + -1 = -0.968245837 Simplifying x + -1 = -0.968245837 Reorder the terms: -1 + x = -0.968245837 Solving -1 + x = -0.968245837 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 = -0.968245837 + 1 Combine like terms: -1 + 1 = 0 0 + x = -0.968245837 + 1 x = -0.968245837 + 1 Combine like terms: -0.968245837 + 1 = 0.031754163 x = 0.031754163 Simplifying x = 0.031754163Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.968245837, 0.031754163}
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