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Simplifying (7 + x)(4 + x) = 100 Multiply (7 + x) * (4 + x) (7(4 + x) + x(4 + x)) = 100 ((4 * 7 + x * 7) + x(4 + x)) = 100 ((28 + 7x) + x(4 + x)) = 100 (28 + 7x + (4 * x + x * x)) = 100 (28 + 7x + (4x + x2)) = 100 Combine like terms: 7x + 4x = 11x (28 + 11x + x2) = 100 Solving 28 + 11x + x2 = 100 Solving for variable 'x'. Reorder the terms: 28 + -100 + 11x + x2 = 100 + -100 Combine like terms: 28 + -100 = -72 -72 + 11x + x2 = 100 + -100 Combine like terms: 100 + -100 = 0 -72 + 11x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '72' to each side of the equation. -72 + 11x + 72 + x2 = 0 + 72 Reorder the terms: -72 + 72 + 11x + x2 = 0 + 72 Combine like terms: -72 + 72 = 0 0 + 11x + x2 = 0 + 72 11x + x2 = 0 + 72 Combine like terms: 0 + 72 = 72 11x + x2 = 72 The x term is 11x. Take half its coefficient (5.5). Square it (30.25) and add it to both sides. Add '30.25' to each side of the equation. 11x + 30.25 + x2 = 72 + 30.25 Reorder the terms: 30.25 + 11x + x2 = 72 + 30.25 Combine like terms: 72 + 30.25 = 102.25 30.25 + 11x + x2 = 102.25 Factor a perfect square on the left side: (x + 5.5)(x + 5.5) = 102.25 Calculate the square root of the right side: 10.111874208 Break this problem into two subproblems by setting (x + 5.5) equal to 10.111874208 and -10.111874208.Subproblem 1
x + 5.5 = 10.111874208 Simplifying x + 5.5 = 10.111874208 Reorder the terms: 5.5 + x = 10.111874208 Solving 5.5 + x = 10.111874208 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + x = 10.111874208 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + x = 10.111874208 + -5.5 x = 10.111874208 + -5.5 Combine like terms: 10.111874208 + -5.5 = 4.611874208 x = 4.611874208 Simplifying x = 4.611874208Subproblem 2
x + 5.5 = -10.111874208 Simplifying x + 5.5 = -10.111874208 Reorder the terms: 5.5 + x = -10.111874208 Solving 5.5 + x = -10.111874208 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + x = -10.111874208 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + x = -10.111874208 + -5.5 x = -10.111874208 + -5.5 Combine like terms: -10.111874208 + -5.5 = -15.611874208 x = -15.611874208 Simplifying x = -15.611874208Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.611874208, -15.611874208}
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