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Simplifying 27x2 = 342x + 72 Reorder the terms: 27x2 = 72 + 342x Solving 27x2 = 72 + 342x Solving for variable 'x'. Reorder the terms: -72 + -342x + 27x2 = 72 + 342x + -72 + -342x Reorder the terms: -72 + -342x + 27x2 = 72 + -72 + 342x + -342x Combine like terms: 72 + -72 = 0 -72 + -342x + 27x2 = 0 + 342x + -342x -72 + -342x + 27x2 = 342x + -342x Combine like terms: 342x + -342x = 0 -72 + -342x + 27x2 = 0 Factor out the Greatest Common Factor (GCF), '9'. 9(-8 + -38x + 3x2) = 0 Ignore the factor 9.Subproblem 1
Set the factor '(-8 + -38x + 3x2)' equal to zero and attempt to solve: Simplifying -8 + -38x + 3x2 = 0 Solving -8 + -38x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2.666666667 + -12.66666667x + x2 = 0 Move the constant term to the right: Add '2.666666667' to each side of the equation. -2.666666667 + -12.66666667x + 2.666666667 + x2 = 0 + 2.666666667 Reorder the terms: -2.666666667 + 2.666666667 + -12.66666667x + x2 = 0 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + -12.66666667x + x2 = 0 + 2.666666667 -12.66666667x + x2 = 0 + 2.666666667 Combine like terms: 0 + 2.666666667 = 2.666666667 -12.66666667x + x2 = 2.666666667 The x term is -12.66666667x. Take half its coefficient (-6.333333335). Square it (40.11111113) and add it to both sides. Add '40.11111113' to each side of the equation. -12.66666667x + 40.11111113 + x2 = 2.666666667 + 40.11111113 Reorder the terms: 40.11111113 + -12.66666667x + x2 = 2.666666667 + 40.11111113 Combine like terms: 2.666666667 + 40.11111113 = 42.777777797 40.11111113 + -12.66666667x + x2 = 42.777777797 Factor a perfect square on the left side: (x + -6.333333335)(x + -6.333333335) = 42.777777797 Calculate the square root of the right side: 6.540472292 Break this problem into two subproblems by setting (x + -6.333333335) equal to 6.540472292 and -6.540472292.Subproblem 1
x + -6.333333335 = 6.540472292 Simplifying x + -6.333333335 = 6.540472292 Reorder the terms: -6.333333335 + x = 6.540472292 Solving -6.333333335 + x = 6.540472292 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.333333335' to each side of the equation. -6.333333335 + 6.333333335 + x = 6.540472292 + 6.333333335 Combine like terms: -6.333333335 + 6.333333335 = 0.000000000 0.000000000 + x = 6.540472292 + 6.333333335 x = 6.540472292 + 6.333333335 Combine like terms: 6.540472292 + 6.333333335 = 12.873805627 x = 12.873805627 Simplifying x = 12.873805627Subproblem 2
x + -6.333333335 = -6.540472292 Simplifying x + -6.333333335 = -6.540472292 Reorder the terms: -6.333333335 + x = -6.540472292 Solving -6.333333335 + x = -6.540472292 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.333333335' to each side of the equation. -6.333333335 + 6.333333335 + x = -6.540472292 + 6.333333335 Combine like terms: -6.333333335 + 6.333333335 = 0.000000000 0.000000000 + x = -6.540472292 + 6.333333335 x = -6.540472292 + 6.333333335 Combine like terms: -6.540472292 + 6.333333335 = -0.207138957 x = -0.207138957 Simplifying x = -0.207138957Solution
The solution to the problem is based on the solutions from the subproblems. x = {12.873805627, -0.207138957}Solution
x = {12.873805627, -0.207138957}
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