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Simplifying x(x + 1) = 511 Reorder the terms: x(1 + x) = 511 (1 * x + x * x) = 511 (1x + x2) = 511 Solving 1x + x2 = 511 Solving for variable 'x'. Reorder the terms: -511 + 1x + x2 = 511 + -511 Combine like terms: 511 + -511 = 0 -511 + 1x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '511' to each side of the equation. -511 + 1x + 511 + x2 = 0 + 511 Reorder the terms: -511 + 511 + 1x + x2 = 0 + 511 Combine like terms: -511 + 511 = 0 0 + 1x + x2 = 0 + 511 1x + x2 = 0 + 511 Combine like terms: 0 + 511 = 511 1x + x2 = 511 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 = 511 + 0.25 Reorder the terms: 0.25 + x + x2 = 511 + 0.25 Combine like terms: 511 + 0.25 = 511.25 0.25 + x + x2 = 511.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 511.25 Calculate the square root of the right side: 22.610838109 Break this problem into two subproblems by setting (x + 0.5) equal to 22.610838109 and -22.610838109.Subproblem 1
x + 0.5 = 22.610838109 Simplifying x + 0.5 = 22.610838109 Reorder the terms: 0.5 + x = 22.610838109 Solving 0.5 + x = 22.610838109 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 = 22.610838109 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 22.610838109 + -0.5 x = 22.610838109 + -0.5 Combine like terms: 22.610838109 + -0.5 = 22.110838109 x = 22.110838109 Simplifying x = 22.110838109Subproblem 2
x + 0.5 = -22.610838109 Simplifying x + 0.5 = -22.610838109 Reorder the terms: 0.5 + x = -22.610838109 Solving 0.5 + x = -22.610838109 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 = -22.610838109 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -22.610838109 + -0.5 x = -22.610838109 + -0.5 Combine like terms: -22.610838109 + -0.5 = -23.110838109 x = -23.110838109 Simplifying x = -23.110838109Solution
The solution to the problem is based on the solutions from the subproblems. x = {22.110838109, -23.110838109}
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