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Simplifying (24 + 2x)(32 + 2x) = 900 Multiply (24 + 2x) * (32 + 2x) (24(32 + 2x) + 2x * (32 + 2x)) = 900 ((32 * 24 + 2x * 24) + 2x * (32 + 2x)) = 900 ((768 + 48x) + 2x * (32 + 2x)) = 900 (768 + 48x + (32 * 2x + 2x * 2x)) = 900 (768 + 48x + (64x + 4x2)) = 900 Combine like terms: 48x + 64x = 112x (768 + 112x + 4x2) = 900 Solving 768 + 112x + 4x2 = 900 Solving for variable 'x'. Reorder the terms: 768 + -900 + 112x + 4x2 = 900 + -900 Combine like terms: 768 + -900 = -132 -132 + 112x + 4x2 = 900 + -900 Combine like terms: 900 + -900 = 0 -132 + 112x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-33 + 28x + x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-33 + 28x + x2)' equal to zero and attempt to solve: Simplifying -33 + 28x + x2 = 0 Solving -33 + 28x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '33' to each side of the equation. -33 + 28x + 33 + x2 = 0 + 33 Reorder the terms: -33 + 33 + 28x + x2 = 0 + 33 Combine like terms: -33 + 33 = 0 0 + 28x + x2 = 0 + 33 28x + x2 = 0 + 33 Combine like terms: 0 + 33 = 33 28x + x2 = 33 The x term is 28x. Take half its coefficient (14). Square it (196) and add it to both sides. Add '196' to each side of the equation. 28x + 196 + x2 = 33 + 196 Reorder the terms: 196 + 28x + x2 = 33 + 196 Combine like terms: 33 + 196 = 229 196 + 28x + x2 = 229 Factor a perfect square on the left side: (x + 14)(x + 14) = 229 Calculate the square root of the right side: 15.13274595 Break this problem into two subproblems by setting (x + 14) equal to 15.13274595 and -15.13274595.Subproblem 1
x + 14 = 15.13274595 Simplifying x + 14 = 15.13274595 Reorder the terms: 14 + x = 15.13274595 Solving 14 + x = 15.13274595 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-14' to each side of the equation. 14 + -14 + x = 15.13274595 + -14 Combine like terms: 14 + -14 = 0 0 + x = 15.13274595 + -14 x = 15.13274595 + -14 Combine like terms: 15.13274595 + -14 = 1.13274595 x = 1.13274595 Simplifying x = 1.13274595Subproblem 2
x + 14 = -15.13274595 Simplifying x + 14 = -15.13274595 Reorder the terms: 14 + x = -15.13274595 Solving 14 + x = -15.13274595 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-14' to each side of the equation. 14 + -14 + x = -15.13274595 + -14 Combine like terms: 14 + -14 = 0 0 + x = -15.13274595 + -14 x = -15.13274595 + -14 Combine like terms: -15.13274595 + -14 = -29.13274595 x = -29.13274595 Simplifying x = -29.13274595Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.13274595, -29.13274595}Solution
x = {1.13274595, -29.13274595}
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