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Simplifying x(x + 1) = 105 Reorder the terms: x(1 + x) = 105 (1 * x + x * x) = 105 (1x + x2) = 105 Solving 1x + x2 = 105 Solving for variable 'x'. Reorder the terms: -105 + 1x + x2 = 105 + -105 Combine like terms: 105 + -105 = 0 -105 + 1x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '105' to each side of the equation. -105 + 1x + 105 + x2 = 0 + 105 Reorder the terms: -105 + 105 + 1x + x2 = 0 + 105 Combine like terms: -105 + 105 = 0 0 + 1x + x2 = 0 + 105 1x + x2 = 0 + 105 Combine like terms: 0 + 105 = 105 1x + x2 = 105 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 = 105 + 0.25 Reorder the terms: 0.25 + x + x2 = 105 + 0.25 Combine like terms: 105 + 0.25 = 105.25 0.25 + x + x2 = 105.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 105.25 Calculate the square root of the right side: 10.259142264 Break this problem into two subproblems by setting (x + 0.5) equal to 10.259142264 and -10.259142264.Subproblem 1
x + 0.5 = 10.259142264 Simplifying x + 0.5 = 10.259142264 Reorder the terms: 0.5 + x = 10.259142264 Solving 0.5 + x = 10.259142264 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 = 10.259142264 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 10.259142264 + -0.5 x = 10.259142264 + -0.5 Combine like terms: 10.259142264 + -0.5 = 9.759142264 x = 9.759142264 Simplifying x = 9.759142264Subproblem 2
x + 0.5 = -10.259142264 Simplifying x + 0.5 = -10.259142264 Reorder the terms: 0.5 + x = -10.259142264 Solving 0.5 + x = -10.259142264 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 = -10.259142264 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -10.259142264 + -0.5 x = -10.259142264 + -0.5 Combine like terms: -10.259142264 + -0.5 = -10.759142264 x = -10.759142264 Simplifying x = -10.759142264Solution
The solution to the problem is based on the solutions from the subproblems. x = {9.759142264, -10.759142264}
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