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Simplifying x + x(x + -2) = 54 Reorder the terms: x + x(-2 + x) = 54 x + (-2 * x + x * x) = 54 x + (-2x + x2) = 54 Combine like terms: x + -2x = -1x -1x + x2 = 54 Solving -1x + x2 = 54 Solving for variable 'x'. Reorder the terms: -54 + -1x + x2 = 54 + -54 Combine like terms: 54 + -54 = 0 -54 + -1x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '54' to each side of the equation. -54 + -1x + 54 + x2 = 0 + 54 Reorder the terms: -54 + 54 + -1x + x2 = 0 + 54 Combine like terms: -54 + 54 = 0 0 + -1x + x2 = 0 + 54 -1x + x2 = 0 + 54 Combine like terms: 0 + 54 = 54 -1x + x2 = 54 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 = 54 + 0.25 Reorder the terms: 0.25 + -1x + x2 = 54 + 0.25 Combine like terms: 54 + 0.25 = 54.25 0.25 + -1x + x2 = 54.25 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 54.25 Calculate the square root of the right side: 7.365459931 Break this problem into two subproblems by setting (x + -0.5) equal to 7.365459931 and -7.365459931.Subproblem 1
x + -0.5 = 7.365459931 Simplifying x + -0.5 = 7.365459931 Reorder the terms: -0.5 + x = 7.365459931 Solving -0.5 + x = 7.365459931 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 = 7.365459931 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 7.365459931 + 0.5 x = 7.365459931 + 0.5 Combine like terms: 7.365459931 + 0.5 = 7.865459931 x = 7.865459931 Simplifying x = 7.865459931Subproblem 2
x + -0.5 = -7.365459931 Simplifying x + -0.5 = -7.365459931 Reorder the terms: -0.5 + x = -7.365459931 Solving -0.5 + x = -7.365459931 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 = -7.365459931 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -7.365459931 + 0.5 x = -7.365459931 + 0.5 Combine like terms: -7.365459931 + 0.5 = -6.865459931 x = -6.865459931 Simplifying x = -6.865459931Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.865459931, -6.865459931}
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