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Simplifying 0 = 8.5x2 + -8.5x + 1 Reorder the terms: 0 = 1 + -8.5x + 8.5x2 Solving 0 = 1 + -8.5x + 8.5x2 Solving for variable 'x'. Combine like terms: 0 + -1 = -1 -1 + 8.5x + -8.5x2 = 1 + -8.5x + 8.5x2 + -1 + 8.5x + -8.5x2 Reorder the terms: -1 + 8.5x + -8.5x2 = 1 + -1 + -8.5x + 8.5x + 8.5x2 + -8.5x2 Combine like terms: 1 + -1 = 0 -1 + 8.5x + -8.5x2 = 0 + -8.5x + 8.5x + 8.5x2 + -8.5x2 -1 + 8.5x + -8.5x2 = -8.5x + 8.5x + 8.5x2 + -8.5x2 Combine like terms: -8.5x + 8.5x = 0.0 -1 + 8.5x + -8.5x2 = 0.0 + 8.5x2 + -8.5x2 -1 + 8.5x + -8.5x2 = 8.5x2 + -8.5x2 Combine like terms: 8.5x2 + -8.5x2 = 0.0 -1 + 8.5x + -8.5x2 = 0.0 Begin completing the square. Divide all terms by -8.5 the coefficient of the squared term: Divide each side by '-8.5'. 0.1176470588 + -1x + x2 = 0 Move the constant term to the right: Add '-0.1176470588' to each side of the equation. 0.1176470588 + -1x + -0.1176470588 + x2 = 0 + -0.1176470588 Reorder the terms: 0.1176470588 + -0.1176470588 + -1x + x2 = 0 + -0.1176470588 Combine like terms: 0.1176470588 + -0.1176470588 = 0.0000000000 0.0000000000 + -1x + x2 = 0 + -0.1176470588 -1x + x2 = 0 + -0.1176470588 Combine like terms: 0 + -0.1176470588 = -0.1176470588 -1x + x2 = -0.1176470588 The x term is -1x. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1x + 0.25 + x2 = -0.1176470588 + 0.25 Reorder the terms: 0.25 + -1x + x2 = -0.1176470588 + 0.25 Combine like terms: -0.1176470588 + 0.25 = 0.1323529412 0.25 + -1x + x2 = 0.1323529412 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 0.1323529412 Calculate the square root of the right side: 0.363803438 Break this problem into two subproblems by setting (x + -0.5) equal to 0.363803438 and -0.363803438.Subproblem 1
x + -0.5 = 0.363803438 Simplifying x + -0.5 = 0.363803438 Reorder the terms: -0.5 + x = 0.363803438 Solving -0.5 + x = 0.363803438 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 0.363803438 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 0.363803438 + 0.5 x = 0.363803438 + 0.5 Combine like terms: 0.363803438 + 0.5 = 0.863803438 x = 0.863803438 Simplifying x = 0.863803438Subproblem 2
x + -0.5 = -0.363803438 Simplifying x + -0.5 = -0.363803438 Reorder the terms: -0.5 + x = -0.363803438 Solving -0.5 + x = -0.363803438 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -0.363803438 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -0.363803438 + 0.5 x = -0.363803438 + 0.5 Combine like terms: -0.363803438 + 0.5 = 0.136196562 x = 0.136196562 Simplifying x = 0.136196562Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.863803438, 0.136196562}
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