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Simplifying -1x2 + 5 = (4x + -1)(4x + -1) Reorder the terms: 5 + -1x2 = (4x + -1)(4x + -1) Reorder the terms: 5 + -1x2 = (-1 + 4x)(4x + -1) Reorder the terms: 5 + -1x2 = (-1 + 4x)(-1 + 4x) Multiply (-1 + 4x) * (-1 + 4x) 5 + -1x2 = (-1(-1 + 4x) + 4x * (-1 + 4x)) 5 + -1x2 = ((-1 * -1 + 4x * -1) + 4x * (-1 + 4x)) 5 + -1x2 = ((1 + -4x) + 4x * (-1 + 4x)) 5 + -1x2 = (1 + -4x + (-1 * 4x + 4x * 4x)) 5 + -1x2 = (1 + -4x + (-4x + 16x2)) Combine like terms: -4x + -4x = -8x 5 + -1x2 = (1 + -8x + 16x2) Solving 5 + -1x2 = 1 + -8x + 16x2 Solving for variable 'x'. Reorder the terms: 5 + -1 + 8x + -1x2 + -16x2 = 1 + -8x + 16x2 + -1 + 8x + -16x2 Combine like terms: 5 + -1 = 4 4 + 8x + -1x2 + -16x2 = 1 + -8x + 16x2 + -1 + 8x + -16x2 Combine like terms: -1x2 + -16x2 = -17x2 4 + 8x + -17x2 = 1 + -8x + 16x2 + -1 + 8x + -16x2 Reorder the terms: 4 + 8x + -17x2 = 1 + -1 + -8x + 8x + 16x2 + -16x2 Combine like terms: 1 + -1 = 0 4 + 8x + -17x2 = 0 + -8x + 8x + 16x2 + -16x2 4 + 8x + -17x2 = -8x + 8x + 16x2 + -16x2 Combine like terms: -8x + 8x = 0 4 + 8x + -17x2 = 0 + 16x2 + -16x2 4 + 8x + -17x2 = 16x2 + -16x2 Combine like terms: 16x2 + -16x2 = 0 4 + 8x + -17x2 = 0 Begin completing the square. Divide all terms by -17 the coefficient of the squared term: Divide each side by '-17'. -0.2352941176 + -0.4705882353x + x2 = 0 Move the constant term to the right: Add '0.2352941176' to each side of the equation. -0.2352941176 + -0.4705882353x + 0.2352941176 + x2 = 0 + 0.2352941176 Reorder the terms: -0.2352941176 + 0.2352941176 + -0.4705882353x + x2 = 0 + 0.2352941176 Combine like terms: -0.2352941176 + 0.2352941176 = 0.0000000000 0.0000000000 + -0.4705882353x + x2 = 0 + 0.2352941176 -0.4705882353x + x2 = 0 + 0.2352941176 Combine like terms: 0 + 0.2352941176 = 0.2352941176 -0.4705882353x + x2 = 0.2352941176 The x term is -0.4705882353x. Take half its coefficient (-0.2352941177). Square it (0.05536332182) and add it to both sides. Add '0.05536332182' to each side of the equation. -0.4705882353x + 0.05536332182 + x2 = 0.2352941176 + 0.05536332182 Reorder the terms: 0.05536332182 + -0.4705882353x + x2 = 0.2352941176 + 0.05536332182 Combine like terms: 0.2352941176 + 0.05536332182 = 0.29065743942 0.05536332182 + -0.4705882353x + x2 = 0.29065743942 Factor a perfect square on the left side: (x + -0.2352941177)(x + -0.2352941177) = 0.29065743942 Calculate the square root of the right side: 0.539126552 Break this problem into two subproblems by setting (x + -0.2352941177) equal to 0.539126552 and -0.539126552.Subproblem 1
x + -0.2352941177 = 0.539126552 Simplifying x + -0.2352941177 = 0.539126552 Reorder the terms: -0.2352941177 + x = 0.539126552 Solving -0.2352941177 + x = 0.539126552 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2352941177' to each side of the equation. -0.2352941177 + 0.2352941177 + x = 0.539126552 + 0.2352941177 Combine like terms: -0.2352941177 + 0.2352941177 = 0.0000000000 0.0000000000 + x = 0.539126552 + 0.2352941177 x = 0.539126552 + 0.2352941177 Combine like terms: 0.539126552 + 0.2352941177 = 0.7744206697 x = 0.7744206697 Simplifying x = 0.7744206697Subproblem 2
x + -0.2352941177 = -0.539126552 Simplifying x + -0.2352941177 = -0.539126552 Reorder the terms: -0.2352941177 + x = -0.539126552 Solving -0.2352941177 + x = -0.539126552 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2352941177' to each side of the equation. -0.2352941177 + 0.2352941177 + x = -0.539126552 + 0.2352941177 Combine like terms: -0.2352941177 + 0.2352941177 = 0.0000000000 0.0000000000 + x = -0.539126552 + 0.2352941177 x = -0.539126552 + 0.2352941177 Combine like terms: -0.539126552 + 0.2352941177 = -0.3038324343 x = -0.3038324343 Simplifying x = -0.3038324343Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.7744206697, -0.3038324343}
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