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Simplifying (20 + 10x)(-8x + 12) = 68 Reorder the terms: (20 + 10x)(12 + -8x) = 68 Multiply (20 + 10x) * (12 + -8x) (20(12 + -8x) + 10x * (12 + -8x)) = 68 ((12 * 20 + -8x * 20) + 10x * (12 + -8x)) = 68 ((240 + -160x) + 10x * (12 + -8x)) = 68 (240 + -160x + (12 * 10x + -8x * 10x)) = 68 (240 + -160x + (120x + -80x2)) = 68 Combine like terms: -160x + 120x = -40x (240 + -40x + -80x2) = 68 Solving 240 + -40x + -80x2 = 68 Solving for variable 'x'. Reorder the terms: 240 + -68 + -40x + -80x2 = 68 + -68 Combine like terms: 240 + -68 = 172 172 + -40x + -80x2 = 68 + -68 Combine like terms: 68 + -68 = 0 172 + -40x + -80x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(43 + -10x + -20x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(43 + -10x + -20x2)' equal to zero and attempt to solve: Simplifying 43 + -10x + -20x2 = 0 Solving 43 + -10x + -20x2 = 0 Begin completing the square. Divide all terms by -20 the coefficient of the squared term: Divide each side by '-20'. -2.15 + 0.5x + x2 = 0.0 Move the constant term to the right: Add '2.15' to each side of the equation. -2.15 + 0.5x + 2.15 + x2 = 0.0 + 2.15 Reorder the terms: -2.15 + 2.15 + 0.5x + x2 = 0.0 + 2.15 Combine like terms: -2.15 + 2.15 = 0.00 0.00 + 0.5x + x2 = 0.0 + 2.15 0.5x + x2 = 0.0 + 2.15 Combine like terms: 0.0 + 2.15 = 2.15 0.5x + x2 = 2.15 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 2.15 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 2.15 + 0.0625 Combine like terms: 2.15 + 0.0625 = 2.2125 0.0625 + 0.5x + x2 = 2.2125 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 2.2125 Calculate the square root of the right side: 1.487447478 Break this problem into two subproblems by setting (x + 0.25) equal to 1.487447478 and -1.487447478.Subproblem 1
x + 0.25 = 1.487447478 Simplifying x + 0.25 = 1.487447478 Reorder the terms: 0.25 + x = 1.487447478 Solving 0.25 + x = 1.487447478 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 1.487447478 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 1.487447478 + -0.25 x = 1.487447478 + -0.25 Combine like terms: 1.487447478 + -0.25 = 1.237447478 x = 1.237447478 Simplifying x = 1.237447478Subproblem 2
x + 0.25 = -1.487447478 Simplifying x + 0.25 = -1.487447478 Reorder the terms: 0.25 + x = -1.487447478 Solving 0.25 + x = -1.487447478 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -1.487447478 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -1.487447478 + -0.25 x = -1.487447478 + -0.25 Combine like terms: -1.487447478 + -0.25 = -1.737447478 x = -1.737447478 Simplifying x = -1.737447478Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.237447478, -1.737447478}Solution
x = {1.237447478, -1.737447478}
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