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Simplifying x = (0.250 + -1x)(0.250 + -1x) Multiply (0.250 + -1x) * (0.250 + -1x) x = (0.250(0.250 + -1x) + -1x * (0.250 + -1x)) x = ((0.250 * 0.250 + -1x * 0.250) + -1x * (0.250 + -1x)) x = ((0.0625 + -0.25x) + -1x * (0.250 + -1x)) x = (0.0625 + -0.25x + (0.250 * -1x + -1x * -1x)) x = (0.0625 + -0.25x + (-0.25x + 1x2)) Combine like terms: -0.25x + -0.25x = -0.5x x = (0.0625 + -0.5x + 1x2) Solving x = 0.0625 + -0.5x + 1x2 Solving for variable 'x'. Reorder the terms: -0.0625 + x + 0.5x + -1x2 = 0.0625 + -0.5x + 1x2 + -0.0625 + 0.5x + -1x2 Combine like terms: x + 0.5x = 1.5x -0.0625 + 1.5x + -1x2 = 0.0625 + -0.5x + 1x2 + -0.0625 + 0.5x + -1x2 Reorder the terms: -0.0625 + 1.5x + -1x2 = 0.0625 + -0.0625 + -0.5x + 0.5x + 1x2 + -1x2 Combine like terms: 0.0625 + -0.0625 = 0.0000 -0.0625 + 1.5x + -1x2 = 0.0000 + -0.5x + 0.5x + 1x2 + -1x2 -0.0625 + 1.5x + -1x2 = -0.5x + 0.5x + 1x2 + -1x2 Combine like terms: -0.5x + 0.5x = 0.0 -0.0625 + 1.5x + -1x2 = 0.0 + 1x2 + -1x2 -0.0625 + 1.5x + -1x2 = 1x2 + -1x2 Combine like terms: 1x2 + -1x2 = 0 -0.0625 + 1.5x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 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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