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Simplifying 16x(x + -1) = 8x + -1 Reorder the terms: 16x(-1 + x) = 8x + -1 (-1 * 16x + x * 16x) = 8x + -1 (-16x + 16x2) = 8x + -1 Reorder the terms: -16x + 16x2 = -1 + 8x Solving -16x + 16x2 = -1 + 8x Solving for variable 'x'. Reorder the terms: 1 + -16x + -8x + 16x2 = -1 + 8x + 1 + -8x Combine like terms: -16x + -8x = -24x 1 + -24x + 16x2 = -1 + 8x + 1 + -8x Reorder the terms: 1 + -24x + 16x2 = -1 + 1 + 8x + -8x Combine like terms: -1 + 1 = 0 1 + -24x + 16x2 = 0 + 8x + -8x 1 + -24x + 16x2 = 8x + -8x Combine like terms: 8x + -8x = 0 1 + -24x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.0625 + -1.5x + x2 = 0 Move the constant term to the right: Add '-0.0625' to each side of the equation. 0.0625 + -1.5x + -0.0625 + x2 = 0 + -0.0625 Reorder the terms: 0.0625 + -0.0625 + -1.5x + x2 = 0 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + -1.5x + x2 = 0 + -0.0625 -1.5x + x2 = 0 + -0.0625 Combine like terms: 0 + -0.0625 = -0.0625 -1.5x + x2 = -0.0625 The x term is -1.5x. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5x + 0.5625 + x2 = -0.0625 + 0.5625 Reorder the terms: 0.5625 + -1.5x + x2 = -0.0625 + 0.5625 Combine like terms: -0.0625 + 0.5625 = 0.5 0.5625 + -1.5x + x2 = 0.5 Factor a perfect square on the left side: (x + -0.75)(x + -0.75) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (x + -0.75) equal to 0.707106781 and -0.707106781.Subproblem 1
x + -0.75 = 0.707106781 Simplifying x + -0.75 = 0.707106781 Reorder the terms: -0.75 + x = 0.707106781 Solving -0.75 + x = 0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 0.707106781 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 0.707106781 + 0.75 x = 0.707106781 + 0.75 Combine like terms: 0.707106781 + 0.75 = 1.457106781 x = 1.457106781 Simplifying x = 1.457106781Subproblem 2
x + -0.75 = -0.707106781 Simplifying x + -0.75 = -0.707106781 Reorder the terms: -0.75 + x = -0.707106781 Solving -0.75 + x = -0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -0.707106781 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -0.707106781 + 0.75 x = -0.707106781 + 0.75 Combine like terms: -0.707106781 + 0.75 = 0.042893219 x = 0.042893219 Simplifying x = 0.042893219Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.457106781, 0.042893219}
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