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Simplifying 26x2 + 96x + 78 = 0 Reorder the terms: 78 + 96x + 26x2 = 0 Solving 78 + 96x + 26x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(39 + 48x + 13x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(39 + 48x + 13x2)' equal to zero and attempt to solve: Simplifying 39 + 48x + 13x2 = 0 Solving 39 + 48x + 13x2 = 0 Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. 3 + 3.692307692x + x2 = 0 Move the constant term to the right: Add '-3' to each side of the equation. 3 + 3.692307692x + -3 + x2 = 0 + -3 Reorder the terms: 3 + -3 + 3.692307692x + x2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + 3.692307692x + x2 = 0 + -3 3.692307692x + x2 = 0 + -3 Combine like terms: 0 + -3 = -3 3.692307692x + x2 = -3 The x term is 3.692307692x. Take half its coefficient (1.846153846). Square it (3.408284023) and add it to both sides. Add '3.408284023' to each side of the equation. 3.692307692x + 3.408284023 + x2 = -3 + 3.408284023 Reorder the terms: 3.408284023 + 3.692307692x + x2 = -3 + 3.408284023 Combine like terms: -3 + 3.408284023 = 0.408284023 3.408284023 + 3.692307692x + x2 = 0.408284023 Factor a perfect square on the left side: (x + 1.846153846)(x + 1.846153846) = 0.408284023 Calculate the square root of the right side: 0.638971066 Break this problem into two subproblems by setting (x + 1.846153846) equal to 0.638971066 and -0.638971066.Subproblem 1
x + 1.846153846 = 0.638971066 Simplifying x + 1.846153846 = 0.638971066 Reorder the terms: 1.846153846 + x = 0.638971066 Solving 1.846153846 + x = 0.638971066 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.846153846' to each side of the equation. 1.846153846 + -1.846153846 + x = 0.638971066 + -1.846153846 Combine like terms: 1.846153846 + -1.846153846 = 0.000000000 0.000000000 + x = 0.638971066 + -1.846153846 x = 0.638971066 + -1.846153846 Combine like terms: 0.638971066 + -1.846153846 = -1.20718278 x = -1.20718278 Simplifying x = -1.20718278Subproblem 2
x + 1.846153846 = -0.638971066 Simplifying x + 1.846153846 = -0.638971066 Reorder the terms: 1.846153846 + x = -0.638971066 Solving 1.846153846 + x = -0.638971066 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.846153846' to each side of the equation. 1.846153846 + -1.846153846 + x = -0.638971066 + -1.846153846 Combine like terms: 1.846153846 + -1.846153846 = 0.000000000 0.000000000 + x = -0.638971066 + -1.846153846 x = -0.638971066 + -1.846153846 Combine like terms: -0.638971066 + -1.846153846 = -2.485124912 x = -2.485124912 Simplifying x = -2.485124912Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.20718278, -2.485124912}Solution
x = {-1.20718278, -2.485124912}
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