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Simplifying (80 + -2x)(40 + -2x) = 1 Multiply (80 + -2x) * (40 + -2x) (80(40 + -2x) + -2x * (40 + -2x)) = 1 ((40 * 80 + -2x * 80) + -2x * (40 + -2x)) = 1 ((3200 + -160x) + -2x * (40 + -2x)) = 1 (3200 + -160x + (40 * -2x + -2x * -2x)) = 1 (3200 + -160x + (-80x + 4x2)) = 1 Combine like terms: -160x + -80x = -240x (3200 + -240x + 4x2) = 1 Solving 3200 + -240x + 4x2 = 1 Solving for variable 'x'. Reorder the terms: 3200 + -1 + -240x + 4x2 = 1 + -1 Combine like terms: 3200 + -1 = 3199 3199 + -240x + 4x2 = 1 + -1 Combine like terms: 1 + -1 = 0 3199 + -240x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 799.75 + -60x + x2 = 0 Move the constant term to the right: Add '-799.75' to each side of the equation. 799.75 + -60x + -799.75 + x2 = 0 + -799.75 Reorder the terms: 799.75 + -799.75 + -60x + x2 = 0 + -799.75 Combine like terms: 799.75 + -799.75 = 0.00 0.00 + -60x + x2 = 0 + -799.75 -60x + x2 = 0 + -799.75 Combine like terms: 0 + -799.75 = -799.75 -60x + x2 = -799.75 The x term is -60x. Take half its coefficient (-30). Square it (900) and add it to both sides. Add '900' to each side of the equation. -60x + 900 + x2 = -799.75 + 900 Reorder the terms: 900 + -60x + x2 = -799.75 + 900 Combine like terms: -799.75 + 900 = 100.25 900 + -60x + x2 = 100.25 Factor a perfect square on the left side: (x + -30)(x + -30) = 100.25 Calculate the square root of the right side: 10.012492197 Break this problem into two subproblems by setting (x + -30) equal to 10.012492197 and -10.012492197.Subproblem 1
x + -30 = 10.012492197 Simplifying x + -30 = 10.012492197 Reorder the terms: -30 + x = 10.012492197 Solving -30 + x = 10.012492197 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '30' to each side of the equation. -30 + 30 + x = 10.012492197 + 30 Combine like terms: -30 + 30 = 0 0 + x = 10.012492197 + 30 x = 10.012492197 + 30 Combine like terms: 10.012492197 + 30 = 40.012492197 x = 40.012492197 Simplifying x = 40.012492197Subproblem 2
x + -30 = -10.012492197 Simplifying x + -30 = -10.012492197 Reorder the terms: -30 + x = -10.012492197 Solving -30 + x = -10.012492197 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '30' to each side of the equation. -30 + 30 + x = -10.012492197 + 30 Combine like terms: -30 + 30 = 0 0 + x = -10.012492197 + 30 x = -10.012492197 + 30 Combine like terms: -10.012492197 + 30 = 19.987507803 x = 19.987507803 Simplifying x = 19.987507803Solution
The solution to the problem is based on the solutions from the subproblems. x = {40.012492197, 19.987507803}
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