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Simplifying (40 + -2x)(80 + -2x) = 1000 Multiply (40 + -2x) * (80 + -2x) (40(80 + -2x) + -2x * (80 + -2x)) = 1000 ((80 * 40 + -2x * 40) + -2x * (80 + -2x)) = 1000 ((3200 + -80x) + -2x * (80 + -2x)) = 1000 (3200 + -80x + (80 * -2x + -2x * -2x)) = 1000 (3200 + -80x + (-160x + 4x2)) = 1000 Combine like terms: -80x + -160x = -240x (3200 + -240x + 4x2) = 1000 Solving 3200 + -240x + 4x2 = 1000 Solving for variable 'x'. Reorder the terms: 3200 + -1000 + -240x + 4x2 = 1000 + -1000 Combine like terms: 3200 + -1000 = 2200 2200 + -240x + 4x2 = 1000 + -1000 Combine like terms: 1000 + -1000 = 0 2200 + -240x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(550 + -60x + x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(550 + -60x + x2)' equal to zero and attempt to solve: Simplifying 550 + -60x + x2 = 0 Solving 550 + -60x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '-550' to each side of the equation. 550 + -60x + -550 + x2 = 0 + -550 Reorder the terms: 550 + -550 + -60x + x2 = 0 + -550 Combine like terms: 550 + -550 = 0 0 + -60x + x2 = 0 + -550 -60x + x2 = 0 + -550 Combine like terms: 0 + -550 = -550 -60x + x2 = -550 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 = -550 + 900 Reorder the terms: 900 + -60x + x2 = -550 + 900 Combine like terms: -550 + 900 = 350 900 + -60x + x2 = 350 Factor a perfect square on the left side: (x + -30)(x + -30) = 350 Calculate the square root of the right side: 18.708286934 Break this problem into two subproblems by setting (x + -30) equal to 18.708286934 and -18.708286934.Subproblem 1
x + -30 = 18.708286934 Simplifying x + -30 = 18.708286934 Reorder the terms: -30 + x = 18.708286934 Solving -30 + x = 18.708286934 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 = 18.708286934 + 30 Combine like terms: -30 + 30 = 0 0 + x = 18.708286934 + 30 x = 18.708286934 + 30 Combine like terms: 18.708286934 + 30 = 48.708286934 x = 48.708286934 Simplifying x = 48.708286934Subproblem 2
x + -30 = -18.708286934 Simplifying x + -30 = -18.708286934 Reorder the terms: -30 + x = -18.708286934 Solving -30 + x = -18.708286934 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 = -18.708286934 + 30 Combine like terms: -30 + 30 = 0 0 + x = -18.708286934 + 30 x = -18.708286934 + 30 Combine like terms: -18.708286934 + 30 = 11.291713066 x = 11.291713066 Simplifying x = 11.291713066Solution
The solution to the problem is based on the solutions from the subproblems. x = {48.708286934, 11.291713066}Solution
x = {48.708286934, 11.291713066}
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