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Simplifying 8x(x + -2) = 1 + 8x Reorder the terms: 8x(-2 + x) = 1 + 8x (-2 * 8x + x * 8x) = 1 + 8x (-16x + 8x2) = 1 + 8x Solving -16x + 8x2 = 1 + 8x Solving for variable 'x'. Reorder the terms: -1 + -16x + -8x + 8x2 = 1 + 8x + -1 + -8x Combine like terms: -16x + -8x = -24x -1 + -24x + 8x2 = 1 + 8x + -1 + -8x Reorder the terms: -1 + -24x + 8x2 = 1 + -1 + 8x + -8x Combine like terms: 1 + -1 = 0 -1 + -24x + 8x2 = 0 + 8x + -8x -1 + -24x + 8x2 = 8x + -8x Combine like terms: 8x + -8x = 0 -1 + -24x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.125 + -3x + x2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + -3x + 0.125 + x2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + -3x + x2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + -3x + x2 = 0 + 0.125 -3x + x2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 -3x + x2 = 0.125 The x term is -3x. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3x + 2.25 + x2 = 0.125 + 2.25 Reorder the terms: 2.25 + -3x + x2 = 0.125 + 2.25 Combine like terms: 0.125 + 2.25 = 2.375 2.25 + -3x + x2 = 2.375 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 2.375 Calculate the square root of the right side: 1.541103501 Break this problem into two subproblems by setting (x + -1.5) equal to 1.541103501 and -1.541103501.Subproblem 1
x + -1.5 = 1.541103501 Simplifying x + -1.5 = 1.541103501 Reorder the terms: -1.5 + x = 1.541103501 Solving -1.5 + x = 1.541103501 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 1.541103501 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 1.541103501 + 1.5 x = 1.541103501 + 1.5 Combine like terms: 1.541103501 + 1.5 = 3.041103501 x = 3.041103501 Simplifying x = 3.041103501Subproblem 2
x + -1.5 = -1.541103501 Simplifying x + -1.5 = -1.541103501 Reorder the terms: -1.5 + x = -1.541103501 Solving -1.5 + x = -1.541103501 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -1.541103501 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -1.541103501 + 1.5 x = -1.541103501 + 1.5 Combine like terms: -1.541103501 + 1.5 = -0.041103501 x = -0.041103501 Simplifying x = -0.041103501Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.041103501, -0.041103501}
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