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Simplifying 8 + x(8 + -1x) + -4x(-2x + 6) = 0 8 + (8 * x + -1x * x) + -4x(-2x + 6) = 0 8 + (8x + -1x2) + -4x(-2x + 6) = 0 Reorder the terms: 8 + 8x + -1x2 + -4x(6 + -2x) = 0 8 + 8x + -1x2 + (6 * -4x + -2x * -4x) = 0 8 + 8x + -1x2 + (-24x + 8x2) = 0 Reorder the terms: 8 + 8x + -24x + -1x2 + 8x2 = 0 Combine like terms: 8x + -24x = -16x 8 + -16x + -1x2 + 8x2 = 0 Combine like terms: -1x2 + 8x2 = 7x2 8 + -16x + 7x2 = 0 Solving 8 + -16x + 7x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 1.142857143 + -2.285714286x + x2 = 0 Move the constant term to the right: Add '-1.142857143' to each side of the equation. 1.142857143 + -2.285714286x + -1.142857143 + x2 = 0 + -1.142857143 Reorder the terms: 1.142857143 + -1.142857143 + -2.285714286x + x2 = 0 + -1.142857143 Combine like terms: 1.142857143 + -1.142857143 = 0.000000000 0.000000000 + -2.285714286x + x2 = 0 + -1.142857143 -2.285714286x + x2 = 0 + -1.142857143 Combine like terms: 0 + -1.142857143 = -1.142857143 -2.285714286x + x2 = -1.142857143 The x term is -2.285714286x. Take half its coefficient (-1.142857143). Square it (1.306122449) and add it to both sides. Add '1.306122449' to each side of the equation. -2.285714286x + 1.306122449 + x2 = -1.142857143 + 1.306122449 Reorder the terms: 1.306122449 + -2.285714286x + x2 = -1.142857143 + 1.306122449 Combine like terms: -1.142857143 + 1.306122449 = 0.163265306 1.306122449 + -2.285714286x + x2 = 0.163265306 Factor a perfect square on the left side: (x + -1.142857143)(x + -1.142857143) = 0.163265306 Calculate the square root of the right side: 0.404061018 Break this problem into two subproblems by setting (x + -1.142857143) equal to 0.404061018 and -0.404061018.Subproblem 1
x + -1.142857143 = 0.404061018 Simplifying x + -1.142857143 = 0.404061018 Reorder the terms: -1.142857143 + x = 0.404061018 Solving -1.142857143 + x = 0.404061018 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + x = 0.404061018 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + x = 0.404061018 + 1.142857143 x = 0.404061018 + 1.142857143 Combine like terms: 0.404061018 + 1.142857143 = 1.546918161 x = 1.546918161 Simplifying x = 1.546918161Subproblem 2
x + -1.142857143 = -0.404061018 Simplifying x + -1.142857143 = -0.404061018 Reorder the terms: -1.142857143 + x = -0.404061018 Solving -1.142857143 + x = -0.404061018 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + x = -0.404061018 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + x = -0.404061018 + 1.142857143 x = -0.404061018 + 1.142857143 Combine like terms: -0.404061018 + 1.142857143 = 0.738796125 x = 0.738796125 Simplifying x = 0.738796125Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.546918161, 0.738796125}
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