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Simplifying (100 + x)(150 + x) = 18000 Multiply (100 + x) * (150 + x) (100(150 + x) + x(150 + x)) = 18000 ((150 * 100 + x * 100) + x(150 + x)) = 18000 ((15000 + 100x) + x(150 + x)) = 18000 (15000 + 100x + (150 * x + x * x)) = 18000 (15000 + 100x + (150x + x2)) = 18000 Combine like terms: 100x + 150x = 250x (15000 + 250x + x2) = 18000 Solving 15000 + 250x + x2 = 18000 Solving for variable 'x'. Reorder the terms: 15000 + -18000 + 250x + x2 = 18000 + -18000 Combine like terms: 15000 + -18000 = -3000 -3000 + 250x + x2 = 18000 + -18000 Combine like terms: 18000 + -18000 = 0 -3000 + 250x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '3000' to each side of the equation. -3000 + 250x + 3000 + x2 = 0 + 3000 Reorder the terms: -3000 + 3000 + 250x + x2 = 0 + 3000 Combine like terms: -3000 + 3000 = 0 0 + 250x + x2 = 0 + 3000 250x + x2 = 0 + 3000 Combine like terms: 0 + 3000 = 3000 250x + x2 = 3000 The x term is 250x. Take half its coefficient (125). Square it (15625) and add it to both sides. Add '15625' to each side of the equation. 250x + 15625 + x2 = 3000 + 15625 Reorder the terms: 15625 + 250x + x2 = 3000 + 15625 Combine like terms: 3000 + 15625 = 18625 15625 + 250x + x2 = 18625 Factor a perfect square on the left side: (x + 125)(x + 125) = 18625 Calculate the square root of the right side: 136.47344064 Break this problem into two subproblems by setting (x + 125) equal to 136.47344064 and -136.47344064.Subproblem 1
x + 125 = 136.47344064 Simplifying x + 125 = 136.47344064 Reorder the terms: 125 + x = 136.47344064 Solving 125 + x = 136.47344064 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-125' to each side of the equation. 125 + -125 + x = 136.47344064 + -125 Combine like terms: 125 + -125 = 0 0 + x = 136.47344064 + -125 x = 136.47344064 + -125 Combine like terms: 136.47344064 + -125 = 11.47344064 x = 11.47344064 Simplifying x = 11.47344064Subproblem 2
x + 125 = -136.47344064 Simplifying x + 125 = -136.47344064 Reorder the terms: 125 + x = -136.47344064 Solving 125 + x = -136.47344064 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-125' to each side of the equation. 125 + -125 + x = -136.47344064 + -125 Combine like terms: 125 + -125 = 0 0 + x = -136.47344064 + -125 x = -136.47344064 + -125 Combine like terms: -136.47344064 + -125 = -261.47344064 x = -261.47344064 Simplifying x = -261.47344064Solution
The solution to the problem is based on the solutions from the subproblems. x = {11.47344064, -261.47344064}
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