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Simplifying x(1 + x) = 1.05 (1 * x + x * x) = 1.05 (1x + x2) = 1.05 Solving 1x + x2 = 1.05 Solving for variable 'x'. Reorder the terms: -1.05 + 1x + x2 = 1.05 + -1.05 Combine like terms: 1.05 + -1.05 = 0.00 -1.05 + 1x + x2 = 0.00 Begin completing the square. Move the constant term to the right: Add '1.05' to each side of the equation. -1.05 + 1x + 1.05 + x2 = 0.00 + 1.05 Reorder the terms: -1.05 + 1.05 + 1x + x2 = 0.00 + 1.05 Combine like terms: -1.05 + 1.05 = 0.00 0.00 + 1x + x2 = 0.00 + 1.05 1x + x2 = 0.00 + 1.05 Combine like terms: 0.00 + 1.05 = 1.05 1x + x2 = 1.05 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 = 1.05 + 0.25 Reorder the terms: 0.25 + x + x2 = 1.05 + 0.25 Combine like terms: 1.05 + 0.25 = 1.3 0.25 + x + x2 = 1.3 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 1.3 Calculate the square root of the right side: 1.140175425 Break this problem into two subproblems by setting (x + 0.5) equal to 1.140175425 and -1.140175425.Subproblem 1
x + 0.5 = 1.140175425 Simplifying x + 0.5 = 1.140175425 Reorder the terms: 0.5 + x = 1.140175425 Solving 0.5 + x = 1.140175425 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 = 1.140175425 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1.140175425 + -0.5 x = 1.140175425 + -0.5 Combine like terms: 1.140175425 + -0.5 = 0.640175425 x = 0.640175425 Simplifying x = 0.640175425Subproblem 2
x + 0.5 = -1.140175425 Simplifying x + 0.5 = -1.140175425 Reorder the terms: 0.5 + x = -1.140175425 Solving 0.5 + x = -1.140175425 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 = -1.140175425 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1.140175425 + -0.5 x = -1.140175425 + -0.5 Combine like terms: -1.140175425 + -0.5 = -1.640175425 x = -1.640175425 Simplifying x = -1.640175425Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.640175425, -1.640175425}
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