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Simplifying 0.000000014 = x(0.1 + 2x) 0.000000014 = (0.1 * x + 2x * x) 0.000000014 = (0.1x + 2x2) Solving 0.000000014 = 0.1x + 2x2 Solving for variable 'x'. Reorder the terms: 0.000000014 + -0.1x + -2x2 = 0.1x + -0.1x + 2x2 + -2x2 Combine like terms: 0.1x + -0.1x = 0.0 0.000000014 + -0.1x + -2x2 = 0.0 + 2x2 + -2x2 0.000000014 + -0.1x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 0.000000014 + -0.1x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -0.000000007 + 0.05x + x2 = 0 Move the constant term to the right: Add '0.000000007' to each side of the equation. -0.000000007 + 0.05x + 0.000000007 + x2 = 0 + 0.000000007 Reorder the terms: -0.000000007 + 0.000000007 + 0.05x + x2 = 0 + 0.000000007 Combine like terms: -0.000000007 + 0.000000007 = 0.000000000 0.000000000 + 0.05x + x2 = 0 + 0.000000007 0.05x + x2 = 0 + 0.000000007 Combine like terms: 0 + 0.000000007 = 0.000000007 0.05x + x2 = 0.000000007 The x term is 0.05x. Take half its coefficient (0.025). Square it (0.000625) and add it to both sides. Add '0.000625' to each side of the equation. 0.05x + 0.000625 + x2 = 0.000000007 + 0.000625 Reorder the terms: 0.000625 + 0.05x + x2 = 0.000000007 + 0.000625 Combine like terms: 0.000000007 + 0.000625 = 0.000625007 0.000625 + 0.05x + x2 = 0.000625007 Factor a perfect square on the left side: (x + 0.025)(x + 0.025) = 0.000625007 Calculate the square root of the right side: 0.02500014 Break this problem into two subproblems by setting (x + 0.025) equal to 0.02500014 and -0.02500014.Subproblem 1
x + 0.025 = 0.02500014 Simplifying x + 0.025 = 0.02500014 Reorder the terms: 0.025 + x = 0.02500014 Solving 0.025 + x = 0.02500014 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.025' to each side of the equation. 0.025 + -0.025 + x = 0.02500014 + -0.025 Combine like terms: 0.025 + -0.025 = 0.000 0.000 + x = 0.02500014 + -0.025 x = 0.02500014 + -0.025 Combine like terms: 0.02500014 + -0.025 = 0.00000014 x = 0.00000014 Simplifying x = 0.00000014Subproblem 2
x + 0.025 = -0.02500014 Simplifying x + 0.025 = -0.02500014 Reorder the terms: 0.025 + x = -0.02500014 Solving 0.025 + x = -0.02500014 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.025' to each side of the equation. 0.025 + -0.025 + x = -0.02500014 + -0.025 Combine like terms: 0.025 + -0.025 = 0.000 0.000 + x = -0.02500014 + -0.025 x = -0.02500014 + -0.025 Combine like terms: -0.02500014 + -0.025 = -0.05000014 x = -0.05000014 Simplifying x = -0.05000014Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.00000014, -0.05000014}
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