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Simplifying 90(x + 13)(x + 23) = 180 Reorder the terms: 90(13 + x)(x + 23) = 180 Reorder the terms: 90(13 + x)(23 + x) = 180 Multiply (13 + x) * (23 + x) 90(13(23 + x) + x(23 + x)) = 180 90((23 * 13 + x * 13) + x(23 + x)) = 180 90((299 + 13x) + x(23 + x)) = 180 90(299 + 13x + (23 * x + x * x)) = 180 90(299 + 13x + (23x + x2)) = 180 Combine like terms: 13x + 23x = 36x 90(299 + 36x + x2) = 180 (299 * 90 + 36x * 90 + x2 * 90) = 180 (26910 + 3240x + 90x2) = 180 Solving 26910 + 3240x + 90x2 = 180 Solving for variable 'x'. Reorder the terms: 26910 + -180 + 3240x + 90x2 = 180 + -180 Combine like terms: 26910 + -180 = 26730 26730 + 3240x + 90x2 = 180 + -180 Combine like terms: 180 + -180 = 0 26730 + 3240x + 90x2 = 0 Factor out the Greatest Common Factor (GCF), '90'. 90(297 + 36x + x2) = 0 Ignore the factor 90.Subproblem 1
Set the factor '(297 + 36x + x2)' equal to zero and attempt to solve: Simplifying 297 + 36x + x2 = 0 Solving 297 + 36x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '-297' to each side of the equation. 297 + 36x + -297 + x2 = 0 + -297 Reorder the terms: 297 + -297 + 36x + x2 = 0 + -297 Combine like terms: 297 + -297 = 0 0 + 36x + x2 = 0 + -297 36x + x2 = 0 + -297 Combine like terms: 0 + -297 = -297 36x + x2 = -297 The x term is 36x. Take half its coefficient (18). Square it (324) and add it to both sides. Add '324' to each side of the equation. 36x + 324 + x2 = -297 + 324 Reorder the terms: 324 + 36x + x2 = -297 + 324 Combine like terms: -297 + 324 = 27 324 + 36x + x2 = 27 Factor a perfect square on the left side: (x + 18)(x + 18) = 27 Calculate the square root of the right side: 5.196152423 Break this problem into two subproblems by setting (x + 18) equal to 5.196152423 and -5.196152423.Subproblem 1
x + 18 = 5.196152423 Simplifying x + 18 = 5.196152423 Reorder the terms: 18 + x = 5.196152423 Solving 18 + x = 5.196152423 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-18' to each side of the equation. 18 + -18 + x = 5.196152423 + -18 Combine like terms: 18 + -18 = 0 0 + x = 5.196152423 + -18 x = 5.196152423 + -18 Combine like terms: 5.196152423 + -18 = -12.803847577 x = -12.803847577 Simplifying x = -12.803847577Subproblem 2
x + 18 = -5.196152423 Simplifying x + 18 = -5.196152423 Reorder the terms: 18 + x = -5.196152423 Solving 18 + x = -5.196152423 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-18' to each side of the equation. 18 + -18 + x = -5.196152423 + -18 Combine like terms: 18 + -18 = 0 0 + x = -5.196152423 + -18 x = -5.196152423 + -18 Combine like terms: -5.196152423 + -18 = -23.196152423 x = -23.196152423 Simplifying x = -23.196152423Solution
The solution to the problem is based on the solutions from the subproblems. x = {-12.803847577, -23.196152423}Solution
x = {-12.803847577, -23.196152423}
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