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Simplifying (14x + 70)(20x + 8) = 180 Reorder the terms: (70 + 14x)(20x + 8) = 180 Reorder the terms: (70 + 14x)(8 + 20x) = 180 Multiply (70 + 14x) * (8 + 20x) (70(8 + 20x) + 14x * (8 + 20x)) = 180 ((8 * 70 + 20x * 70) + 14x * (8 + 20x)) = 180 ((560 + 1400x) + 14x * (8 + 20x)) = 180 (560 + 1400x + (8 * 14x + 20x * 14x)) = 180 (560 + 1400x + (112x + 280x2)) = 180 Combine like terms: 1400x + 112x = 1512x (560 + 1512x + 280x2) = 180 Solving 560 + 1512x + 280x2 = 180 Solving for variable 'x'. Reorder the terms: 560 + -180 + 1512x + 280x2 = 180 + -180 Combine like terms: 560 + -180 = 380 380 + 1512x + 280x2 = 180 + -180 Combine like terms: 180 + -180 = 0 380 + 1512x + 280x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(95 + 378x + 70x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(95 + 378x + 70x2)' equal to zero and attempt to solve: Simplifying 95 + 378x + 70x2 = 0 Solving 95 + 378x + 70x2 = 0 Begin completing the square. Divide all terms by 70 the coefficient of the squared term: Divide each side by '70'. 1.357142857 + 5.4x + x2 = 0.0 Move the constant term to the right: Add '-1.357142857' to each side of the equation. 1.357142857 + 5.4x + -1.357142857 + x2 = 0.0 + -1.357142857 Reorder the terms: 1.357142857 + -1.357142857 + 5.4x + x2 = 0.0 + -1.357142857 Combine like terms: 1.357142857 + -1.357142857 = 0.000000000 0.000000000 + 5.4x + x2 = 0.0 + -1.357142857 5.4x + x2 = 0.0 + -1.357142857 Combine like terms: 0.0 + -1.357142857 = -1.357142857 5.4x + x2 = -1.357142857 The x term is 5.4x. Take half its coefficient (2.7). Square it (7.29) and add it to both sides. Add '7.29' to each side of the equation. 5.4x + 7.29 + x2 = -1.357142857 + 7.29 Reorder the terms: 7.29 + 5.4x + x2 = -1.357142857 + 7.29 Combine like terms: -1.357142857 + 7.29 = 5.932857143 7.29 + 5.4x + x2 = 5.932857143 Factor a perfect square on the left side: (x + 2.7)(x + 2.7) = 5.932857143 Calculate the square root of the right side: 2.435745706 Break this problem into two subproblems by setting (x + 2.7) equal to 2.435745706 and -2.435745706.Subproblem 1
x + 2.7 = 2.435745706 Simplifying x + 2.7 = 2.435745706 Reorder the terms: 2.7 + x = 2.435745706 Solving 2.7 + x = 2.435745706 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.7' to each side of the equation. 2.7 + -2.7 + x = 2.435745706 + -2.7 Combine like terms: 2.7 + -2.7 = 0.0 0.0 + x = 2.435745706 + -2.7 x = 2.435745706 + -2.7 Combine like terms: 2.435745706 + -2.7 = -0.264254294 x = -0.264254294 Simplifying x = -0.264254294Subproblem 2
x + 2.7 = -2.435745706 Simplifying x + 2.7 = -2.435745706 Reorder the terms: 2.7 + x = -2.435745706 Solving 2.7 + x = -2.435745706 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.7' to each side of the equation. 2.7 + -2.7 + x = -2.435745706 + -2.7 Combine like terms: 2.7 + -2.7 = 0.0 0.0 + x = -2.435745706 + -2.7 x = -2.435745706 + -2.7 Combine like terms: -2.435745706 + -2.7 = -5.135745706 x = -5.135745706 Simplifying x = -5.135745706Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.264254294, -5.135745706}Solution
x = {-0.264254294, -5.135745706}
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