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Simplifying 88x2 + 84x + 20 = 180 Reorder the terms: 20 + 84x + 88x2 = 180 Solving 20 + 84x + 88x2 = 180 Solving for variable 'x'. Reorder the terms: 20 + -180 + 84x + 88x2 = 180 + -180 Combine like terms: 20 + -180 = -160 -160 + 84x + 88x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -160 + 84x + 88x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-40 + 21x + 22x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-40 + 21x + 22x2)' equal to zero and attempt to solve: Simplifying -40 + 21x + 22x2 = 0 Solving -40 + 21x + 22x2 = 0 Begin completing the square. Divide all terms by 22 the coefficient of the squared term: Divide each side by '22'. -1.818181818 + 0.9545454545x + x2 = 0 Move the constant term to the right: Add '1.818181818' to each side of the equation. -1.818181818 + 0.9545454545x + 1.818181818 + x2 = 0 + 1.818181818 Reorder the terms: -1.818181818 + 1.818181818 + 0.9545454545x + x2 = 0 + 1.818181818 Combine like terms: -1.818181818 + 1.818181818 = 0.000000000 0.000000000 + 0.9545454545x + x2 = 0 + 1.818181818 0.9545454545x + x2 = 0 + 1.818181818 Combine like terms: 0 + 1.818181818 = 1.818181818 0.9545454545x + x2 = 1.818181818 The x term is 0.9545454545x. Take half its coefficient (0.4772727273). Square it (0.2277892562) and add it to both sides. Add '0.2277892562' to each side of the equation. 0.9545454545x + 0.2277892562 + x2 = 1.818181818 + 0.2277892562 Reorder the terms: 0.2277892562 + 0.9545454545x + x2 = 1.818181818 + 0.2277892562 Combine like terms: 1.818181818 + 0.2277892562 = 2.0459710742 0.2277892562 + 0.9545454545x + x2 = 2.0459710742 Factor a perfect square on the left side: (x + 0.4772727273)(x + 0.4772727273) = 2.0459710742 Calculate the square root of the right side: 1.430374452 Break this problem into two subproblems by setting (x + 0.4772727273) equal to 1.430374452 and -1.430374452.Subproblem 1
x + 0.4772727273 = 1.430374452 Simplifying x + 0.4772727273 = 1.430374452 Reorder the terms: 0.4772727273 + x = 1.430374452 Solving 0.4772727273 + x = 1.430374452 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4772727273' to each side of the equation. 0.4772727273 + -0.4772727273 + x = 1.430374452 + -0.4772727273 Combine like terms: 0.4772727273 + -0.4772727273 = 0.0000000000 0.0000000000 + x = 1.430374452 + -0.4772727273 x = 1.430374452 + -0.4772727273 Combine like terms: 1.430374452 + -0.4772727273 = 0.9531017247 x = 0.9531017247 Simplifying x = 0.9531017247Subproblem 2
x + 0.4772727273 = -1.430374452 Simplifying x + 0.4772727273 = -1.430374452 Reorder the terms: 0.4772727273 + x = -1.430374452 Solving 0.4772727273 + x = -1.430374452 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4772727273' to each side of the equation. 0.4772727273 + -0.4772727273 + x = -1.430374452 + -0.4772727273 Combine like terms: 0.4772727273 + -0.4772727273 = 0.0000000000 0.0000000000 + x = -1.430374452 + -0.4772727273 x = -1.430374452 + -0.4772727273 Combine like terms: -1.430374452 + -0.4772727273 = -1.9076471793 x = -1.9076471793 Simplifying x = -1.9076471793Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.9531017247, -1.9076471793}Solution
x = {0.9531017247, -1.9076471793}
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