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Simplifying (10 + x)(5 + x)(2) = 150 Reorder the terms for easier multiplication: 2(10 + x)(5 + x) = 150 Multiply (10 + x) * (5 + x) 2(10(5 + x) + x(5 + x)) = 150 2((5 * 10 + x * 10) + x(5 + x)) = 150 2((50 + 10x) + x(5 + x)) = 150 2(50 + 10x + (5 * x + x * x)) = 150 2(50 + 10x + (5x + x2)) = 150 Combine like terms: 10x + 5x = 15x 2(50 + 15x + x2) = 150 (50 * 2 + 15x * 2 + x2 * 2) = 150 (100 + 30x + 2x2) = 150 Solving 100 + 30x + 2x2 = 150 Solving for variable 'x'. Reorder the terms: 100 + -150 + 30x + 2x2 = 150 + -150 Combine like terms: 100 + -150 = -50 -50 + 30x + 2x2 = 150 + -150 Combine like terms: 150 + -150 = 0 -50 + 30x + 2x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-25 + 15x + x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-25 + 15x + x2)' equal to zero and attempt to solve: Simplifying -25 + 15x + x2 = 0 Solving -25 + 15x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '25' to each side of the equation. -25 + 15x + 25 + x2 = 0 + 25 Reorder the terms: -25 + 25 + 15x + x2 = 0 + 25 Combine like terms: -25 + 25 = 0 0 + 15x + x2 = 0 + 25 15x + x2 = 0 + 25 Combine like terms: 0 + 25 = 25 15x + x2 = 25 The x term is 15x. Take half its coefficient (7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. 15x + 56.25 + x2 = 25 + 56.25 Reorder the terms: 56.25 + 15x + x2 = 25 + 56.25 Combine like terms: 25 + 56.25 = 81.25 56.25 + 15x + x2 = 81.25 Factor a perfect square on the left side: (x + 7.5)(x + 7.5) = 81.25 Calculate the square root of the right side: 9.013878189 Break this problem into two subproblems by setting (x + 7.5) equal to 9.013878189 and -9.013878189.Subproblem 1
x + 7.5 = 9.013878189 Simplifying x + 7.5 = 9.013878189 Reorder the terms: 7.5 + x = 9.013878189 Solving 7.5 + x = 9.013878189 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + x = 9.013878189 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + x = 9.013878189 + -7.5 x = 9.013878189 + -7.5 Combine like terms: 9.013878189 + -7.5 = 1.513878189 x = 1.513878189 Simplifying x = 1.513878189Subproblem 2
x + 7.5 = -9.013878189 Simplifying x + 7.5 = -9.013878189 Reorder the terms: 7.5 + x = -9.013878189 Solving 7.5 + x = -9.013878189 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + x = -9.013878189 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + x = -9.013878189 + -7.5 x = -9.013878189 + -7.5 Combine like terms: -9.013878189 + -7.5 = -16.513878189 x = -16.513878189 Simplifying x = -16.513878189Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.513878189, -16.513878189}Solution
x = {1.513878189, -16.513878189}
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