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Simplifying (20 + 40x)(900 + -30x) = x Multiply (20 + 40x) * (900 + -30x) (20(900 + -30x) + 40x * (900 + -30x)) = x ((900 * 20 + -30x * 20) + 40x * (900 + -30x)) = x ((18000 + -600x) + 40x * (900 + -30x)) = x (18000 + -600x + (900 * 40x + -30x * 40x)) = x (18000 + -600x + (36000x + -1200x2)) = x Combine like terms: -600x + 36000x = 35400x (18000 + 35400x + -1200x2) = x Solving 18000 + 35400x + -1200x2 = x Solving for variable 'x'. Reorder the terms: 18000 + 35400x + -1x + -1200x2 = x + -1x Combine like terms: 35400x + -1x = 35399x 18000 + 35399x + -1200x2 = x + -1x Combine like terms: x + -1x = 0 18000 + 35399x + -1200x2 = 0 Begin completing the square. Divide all terms by -1200 the coefficient of the squared term: Divide each side by '-1200'. -15 + -29.49916667x + x2 = 0.00 Move the constant term to the right: Add '15' to each side of the equation. -15 + -29.49916667x + 15 + x2 = 0.00 + 15 Reorder the terms: -15 + 15 + -29.49916667x + x2 = 0.00 + 15 Combine like terms: -15 + 15 = 0 0 + -29.49916667x + x2 = 0.00 + 15 -29.49916667x + x2 = 0.00 + 15 Combine like terms: 0.00 + 15 = 15 -29.49916667x + x2 = 15 The x term is -29.49916667x. Take half its coefficient (-14.74958334). Square it (217.5502087) and add it to both sides. Add '217.5502087' to each side of the equation. -29.49916667x + 217.5502087 + x2 = 15 + 217.5502087 Reorder the terms: 217.5502087 + -29.49916667x + x2 = 15 + 217.5502087 Combine like terms: 15 + 217.5502087 = 232.5502087 217.5502087 + -29.49916667x + x2 = 232.5502087 Factor a perfect square on the left side: (x + -14.74958334)(x + -14.74958334) = 232.5502087 Calculate the square root of the right side: 15.249597001 Break this problem into two subproblems by setting (x + -14.74958334) equal to 15.249597001 and -15.249597001.Subproblem 1
x + -14.74958334 = 15.249597001 Simplifying x + -14.74958334 = 15.249597001 Reorder the terms: -14.74958334 + x = 15.249597001 Solving -14.74958334 + x = 15.249597001 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '14.74958334' to each side of the equation. -14.74958334 + 14.74958334 + x = 15.249597001 + 14.74958334 Combine like terms: -14.74958334 + 14.74958334 = 0.00000000 0.00000000 + x = 15.249597001 + 14.74958334 x = 15.249597001 + 14.74958334 Combine like terms: 15.249597001 + 14.74958334 = 29.999180341 x = 29.999180341 Simplifying x = 29.999180341Subproblem 2
x + -14.74958334 = -15.249597001 Simplifying x + -14.74958334 = -15.249597001 Reorder the terms: -14.74958334 + x = -15.249597001 Solving -14.74958334 + x = -15.249597001 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '14.74958334' to each side of the equation. -14.74958334 + 14.74958334 + x = -15.249597001 + 14.74958334 Combine like terms: -14.74958334 + 14.74958334 = 0.00000000 0.00000000 + x = -15.249597001 + 14.74958334 x = -15.249597001 + 14.74958334 Combine like terms: -15.249597001 + 14.74958334 = -0.500013661 x = -0.500013661 Simplifying x = -0.500013661Solution
The solution to the problem is based on the solutions from the subproblems. x = {29.999180341, -0.500013661}
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