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Simplifying 530.8(1 + -0.1x)(1 + x) = 609.7 Multiply (1 + -0.1x) * (1 + x) 530.8(1(1 + x) + -0.1x * (1 + x)) = 609.7 530.8((1 * 1 + x * 1) + -0.1x * (1 + x)) = 609.7 530.8((1 + 1x) + -0.1x * (1 + x)) = 609.7 530.8(1 + 1x + (1 * -0.1x + x * -0.1x)) = 609.7 530.8(1 + 1x + (-0.1x + -0.1x2)) = 609.7 Combine like terms: 1x + -0.1x = 0.9x 530.8(1 + 0.9x + -0.1x2) = 609.7 (1 * 530.8 + 0.9x * 530.8 + -0.1x2 * 530.8) = 609.7 (530.8 + 477.72x + -53.08x2) = 609.7 Solving 530.8 + 477.72x + -53.08x2 = 609.7 Solving for variable 'x'. Reorder the terms: 530.8 + -609.7 + 477.72x + -53.08x2 = 609.7 + -609.7 Combine like terms: 530.8 + -609.7 = -78.9 -78.9 + 477.72x + -53.08x2 = 609.7 + -609.7 Combine like terms: 609.7 + -609.7 = 0.0 -78.9 + 477.72x + -53.08x2 = 0.0 Begin completing the square. Divide all terms by -53.08 the coefficient of the squared term: Divide each side by '-53.08'. 1.486435569 + -9x + x2 = 0 Move the constant term to the right: Add '-1.486435569' to each side of the equation. 1.486435569 + -9x + -1.486435569 + x2 = 0 + -1.486435569 Reorder the terms: 1.486435569 + -1.486435569 + -9x + x2 = 0 + -1.486435569 Combine like terms: 1.486435569 + -1.486435569 = 0.000000000 0.000000000 + -9x + x2 = 0 + -1.486435569 -9x + x2 = 0 + -1.486435569 Combine like terms: 0 + -1.486435569 = -1.486435569 -9x + x2 = -1.486435569 The x term is -9x. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9x + 20.25 + x2 = -1.486435569 + 20.25 Reorder the terms: 20.25 + -9x + x2 = -1.486435569 + 20.25 Combine like terms: -1.486435569 + 20.25 = 18.763564431 20.25 + -9x + x2 = 18.763564431 Factor a perfect square on the left side: (x + -4.5)(x + -4.5) = 18.763564431 Calculate the square root of the right side: 4.331693021 Break this problem into two subproblems by setting (x + -4.5) equal to 4.331693021 and -4.331693021.Subproblem 1
x + -4.5 = 4.331693021 Simplifying x + -4.5 = 4.331693021 Reorder the terms: -4.5 + x = 4.331693021 Solving -4.5 + x = 4.331693021 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x = 4.331693021 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x = 4.331693021 + 4.5 x = 4.331693021 + 4.5 Combine like terms: 4.331693021 + 4.5 = 8.831693021 x = 8.831693021 Simplifying x = 8.831693021Subproblem 2
x + -4.5 = -4.331693021 Simplifying x + -4.5 = -4.331693021 Reorder the terms: -4.5 + x = -4.331693021 Solving -4.5 + x = -4.331693021 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x = -4.331693021 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x = -4.331693021 + 4.5 x = -4.331693021 + 4.5 Combine like terms: -4.331693021 + 4.5 = 0.168306979 x = 0.168306979 Simplifying x = 0.168306979Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.831693021, 0.168306979}
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