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Simplifying x = -16x2 + 32x + 5 Reorder the terms: x = 5 + 32x + -16x2 Solving x = 5 + 32x + -16x2 Solving for variable 'x'. Reorder the terms: -5 + x + -32x + 16x2 = 5 + 32x + -16x2 + -5 + -32x + 16x2 Combine like terms: x + -32x = -31x -5 + -31x + 16x2 = 5 + 32x + -16x2 + -5 + -32x + 16x2 Reorder the terms: -5 + -31x + 16x2 = 5 + -5 + 32x + -32x + -16x2 + 16x2 Combine like terms: 5 + -5 = 0 -5 + -31x + 16x2 = 0 + 32x + -32x + -16x2 + 16x2 -5 + -31x + 16x2 = 32x + -32x + -16x2 + 16x2 Combine like terms: 32x + -32x = 0 -5 + -31x + 16x2 = 0 + -16x2 + 16x2 -5 + -31x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -5 + -31x + 16x2 = 0 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 = 1.2509765625 0.9384765625 + -1.9375x + x2 = 1.2509765625 Factor a perfect square on the left side: (x + -0.96875)(x + -0.96875) = 1.2509765625 Calculate the square root of the right side: 1.118470636 Break this problem into two subproblems by setting (x + -0.96875) equal to 1.118470636 and -1.118470636.Subproblem 1
x + -0.96875 = 1.118470636 Simplifying x + -0.96875 = 1.118470636 Reorder the terms: -0.96875 + x = 1.118470636 Solving -0.96875 + x = 1.118470636 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 = 1.118470636 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + x = 1.118470636 + 0.96875 x = 1.118470636 + 0.96875 Combine like terms: 1.118470636 + 0.96875 = 2.087220636 x = 2.087220636 Simplifying x = 2.087220636Subproblem 2
x + -0.96875 = -1.118470636 Simplifying x + -0.96875 = -1.118470636 Reorder the terms: -0.96875 + x = -1.118470636 Solving -0.96875 + x = -1.118470636 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 = -1.118470636 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + x = -1.118470636 + 0.96875 x = -1.118470636 + 0.96875 Combine like terms: -1.118470636 + 0.96875 = -0.149720636 x = -0.149720636 Simplifying x = -0.149720636Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.087220636, -0.149720636}
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