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Simplifying x + (2x + 6)(2x + -6) = 180 Reorder the terms: x + (6 + 2x)(2x + -6) = 180 Reorder the terms: x + (6 + 2x)(-6 + 2x) = 180 Multiply (6 + 2x) * (-6 + 2x) x + (6(-6 + 2x) + 2x * (-6 + 2x)) = 180 x + ((-6 * 6 + 2x * 6) + 2x * (-6 + 2x)) = 180 x + ((-36 + 12x) + 2x * (-6 + 2x)) = 180 x + (-36 + 12x + (-6 * 2x + 2x * 2x)) = 180 x + (-36 + 12x + (-12x + 4x2)) = 180 Combine like terms: 12x + -12x = 0 x + (-36 + 0 + 4x2) = 180 x + (-36 + 4x2) = 180 Reorder the terms: -36 + x + 4x2 = 180 Solving -36 + x + 4x2 = 180 Solving for variable 'x'. Reorder the terms: -36 + -180 + x + 4x2 = 180 + -180 Combine like terms: -36 + -180 = -216 -216 + x + 4x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -216 + x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -54 + 0.25x + x2 = 0 Move the constant term to the right: Add '54' to each side of the equation. -54 + 0.25x + 54 + x2 = 0 + 54 Reorder the terms: -54 + 54 + 0.25x + x2 = 0 + 54 Combine like terms: -54 + 54 = 0 0 + 0.25x + x2 = 0 + 54 0.25x + x2 = 0 + 54 Combine like terms: 0 + 54 = 54 0.25x + x2 = 54 The x term is x. 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. 0.25x + 0.25 + x2 = 54 + 0.25 Reorder the terms: 0.25 + 0.25x + x2 = 54 + 0.25 Combine like terms: 54 + 0.25 = 54.25 0.25 + 0.25x + 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 = 6.865459931 x = 6.865459931 Simplifying x = 6.865459931Subproblem 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 = -7.865459931 x = -7.865459931 Simplifying x = -7.865459931Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.865459931, -7.865459931}
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