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Simplifying x(x + 1) = 429 Reorder the terms: x(1 + x) = 429 (1 * x + x * x) = 429 (1x + x2) = 429 Solving 1x + x2 = 429 Solving for variable 'x'. Reorder the terms: -429 + 1x + x2 = 429 + -429 Combine like terms: 429 + -429 = 0 -429 + 1x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '429' to each side of the equation. -429 + 1x + 429 + x2 = 0 + 429 Reorder the terms: -429 + 429 + 1x + x2 = 0 + 429 Combine like terms: -429 + 429 = 0 0 + 1x + x2 = 0 + 429 1x + x2 = 0 + 429 Combine like terms: 0 + 429 = 429 1x + x2 = 429 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 = 429 + 0.25 Reorder the terms: 0.25 + x + x2 = 429 + 0.25 Combine like terms: 429 + 0.25 = 429.25 0.25 + x + x2 = 429.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 429.25 Calculate the square root of the right side: 20.718349355 Break this problem into two subproblems by setting (x + 0.5) equal to 20.718349355 and -20.718349355.Subproblem 1
x + 0.5 = 20.718349355 Simplifying x + 0.5 = 20.718349355 Reorder the terms: 0.5 + x = 20.718349355 Solving 0.5 + x = 20.718349355 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 = 20.718349355 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 20.718349355 + -0.5 x = 20.718349355 + -0.5 Combine like terms: 20.718349355 + -0.5 = 20.218349355 x = 20.218349355 Simplifying x = 20.218349355Subproblem 2
x + 0.5 = -20.718349355 Simplifying x + 0.5 = -20.718349355 Reorder the terms: 0.5 + x = -20.718349355 Solving 0.5 + x = -20.718349355 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 = -20.718349355 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -20.718349355 + -0.5 x = -20.718349355 + -0.5 Combine like terms: -20.718349355 + -0.5 = -21.218349355 x = -21.218349355 Simplifying x = -21.218349355Solution
The solution to the problem is based on the solutions from the subproblems. x = {20.218349355, -21.218349355}
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