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Simplifying X(X + 1) = 138 Reorder the terms: X(1 + X) = 138 (1 * X + X * X) = 138 (1X + X2) = 138 Solving 1X + X2 = 138 Solving for variable 'X'. Reorder the terms: -138 + 1X + X2 = 138 + -138 Combine like terms: 138 + -138 = 0 -138 + 1X + X2 = 0 Begin completing the square. Move the constant term to the right: Add '138' to each side of the equation. -138 + 1X + 138 + X2 = 0 + 138 Reorder the terms: -138 + 138 + 1X + X2 = 0 + 138 Combine like terms: -138 + 138 = 0 0 + 1X + X2 = 0 + 138 1X + X2 = 0 + 138 Combine like terms: 0 + 138 = 138 1X + X2 = 138 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. X + 0.25 + X2 = 138 + 0.25 Reorder the terms: 0.25 + X + X2 = 138 + 0.25 Combine like terms: 138 + 0.25 = 138.25 0.25 + X + X2 = 138.25 Factor a perfect square on the left side: (X + 0.5)(X + 0.5) = 138.25 Calculate the square root of the right side: 11.757976016 Break this problem into two subproblems by setting (X + 0.5) equal to 11.757976016 and -11.757976016.Subproblem 1
X + 0.5 = 11.757976016 Simplifying X + 0.5 = 11.757976016 Reorder the terms: 0.5 + X = 11.757976016 Solving 0.5 + X = 11.757976016 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 = 11.757976016 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + X = 11.757976016 + -0.5 X = 11.757976016 + -0.5 Combine like terms: 11.757976016 + -0.5 = 11.257976016 X = 11.257976016 Simplifying X = 11.257976016Subproblem 2
X + 0.5 = -11.757976016 Simplifying X + 0.5 = -11.757976016 Reorder the terms: 0.5 + X = -11.757976016 Solving 0.5 + X = -11.757976016 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 = -11.757976016 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + X = -11.757976016 + -0.5 X = -11.757976016 + -0.5 Combine like terms: -11.757976016 + -0.5 = -12.257976016 X = -12.257976016 Simplifying X = -12.257976016Solution
The solution to the problem is based on the solutions from the subproblems. X = {11.257976016, -12.257976016}
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