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Simplifying 180 = 45 + x(2x + -2) Reorder the terms: 180 = 45 + x(-2 + 2x) 180 = 45 + (-2 * x + 2x * x) 180 = 45 + (-2x + 2x2) Solving 180 = 45 + -2x + 2x2 Solving for variable 'x'. Combine like terms: 180 + -45 = 135 135 + 2x + -2x2 = 45 + -2x + 2x2 + -45 + 2x + -2x2 Reorder the terms: 135 + 2x + -2x2 = 45 + -45 + -2x + 2x + 2x2 + -2x2 Combine like terms: 45 + -45 = 0 135 + 2x + -2x2 = 0 + -2x + 2x + 2x2 + -2x2 135 + 2x + -2x2 = -2x + 2x + 2x2 + -2x2 Combine like terms: -2x + 2x = 0 135 + 2x + -2x2 = 0 + 2x2 + -2x2 135 + 2x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 135 + 2x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -67.5 + -1x + x2 = 0 Move the constant term to the right: Add '67.5' to each side of the equation. -67.5 + -1x + 67.5 + x2 = 0 + 67.5 Reorder the terms: -67.5 + 67.5 + -1x + x2 = 0 + 67.5 Combine like terms: -67.5 + 67.5 = 0.0 0.0 + -1x + x2 = 0 + 67.5 -1x + x2 = 0 + 67.5 Combine like terms: 0 + 67.5 = 67.5 -1x + x2 = 67.5 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 = 67.5 + 0.25 Reorder the terms: 0.25 + -1x + x2 = 67.5 + 0.25 Combine like terms: 67.5 + 0.25 = 67.75 0.25 + -1x + x2 = 67.75 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 67.75 Calculate the square root of the right side: 8.231038817 Break this problem into two subproblems by setting (x + -0.5) equal to 8.231038817 and -8.231038817.Subproblem 1
x + -0.5 = 8.231038817 Simplifying x + -0.5 = 8.231038817 Reorder the terms: -0.5 + x = 8.231038817 Solving -0.5 + x = 8.231038817 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 = 8.231038817 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 8.231038817 + 0.5 x = 8.231038817 + 0.5 Combine like terms: 8.231038817 + 0.5 = 8.731038817 x = 8.731038817 Simplifying x = 8.731038817Subproblem 2
x + -0.5 = -8.231038817 Simplifying x + -0.5 = -8.231038817 Reorder the terms: -0.5 + x = -8.231038817 Solving -0.5 + x = -8.231038817 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 = -8.231038817 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -8.231038817 + 0.5 x = -8.231038817 + 0.5 Combine like terms: -8.231038817 + 0.5 = -7.731038817 x = -7.731038817 Simplifying x = -7.731038817Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.731038817, -7.731038817}
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