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Simplifying 0 = -36x2 + 192x + -144 Reorder the terms: 0 = -144 + 192x + -36x2 Solving 0 = -144 + 192x + -36x2 Solving for variable 'x'. Combine like terms: 0 + 144 = 144 144 + -192x + 36x2 = -144 + 192x + -36x2 + 144 + -192x + 36x2 Reorder the terms: 144 + -192x + 36x2 = -144 + 144 + 192x + -192x + -36x2 + 36x2 Combine like terms: -144 + 144 = 0 144 + -192x + 36x2 = 0 + 192x + -192x + -36x2 + 36x2 144 + -192x + 36x2 = 192x + -192x + -36x2 + 36x2 Combine like terms: 192x + -192x = 0 144 + -192x + 36x2 = 0 + -36x2 + 36x2 144 + -192x + 36x2 = -36x2 + 36x2 Combine like terms: -36x2 + 36x2 = 0 144 + -192x + 36x2 = 0 Factor out the Greatest Common Factor (GCF), '12'. 12(12 + -16x + 3x2) = 0 Ignore the factor 12.Subproblem 1
Set the factor '(12 + -16x + 3x2)' equal to zero and attempt to solve: Simplifying 12 + -16x + 3x2 = 0 Solving 12 + -16x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 4 + -5.333333333x + x2 = 0 Move the constant term to the right: Add '-4' to each side of the equation. 4 + -5.333333333x + -4 + x2 = 0 + -4 Reorder the terms: 4 + -4 + -5.333333333x + x2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -5.333333333x + x2 = 0 + -4 -5.333333333x + x2 = 0 + -4 Combine like terms: 0 + -4 = -4 -5.333333333x + x2 = -4 The x term is -5.333333333x. Take half its coefficient (-2.666666667). Square it (7.111111113) and add it to both sides. Add '7.111111113' to each side of the equation. -5.333333333x + 7.111111113 + x2 = -4 + 7.111111113 Reorder the terms: 7.111111113 + -5.333333333x + x2 = -4 + 7.111111113 Combine like terms: -4 + 7.111111113 = 3.111111113 7.111111113 + -5.333333333x + x2 = 3.111111113 Factor a perfect square on the left side: (x + -2.666666667)(x + -2.666666667) = 3.111111113 Calculate the square root of the right side: 1.763834208 Break this problem into two subproblems by setting (x + -2.666666667) equal to 1.763834208 and -1.763834208.Subproblem 1
x + -2.666666667 = 1.763834208 Simplifying x + -2.666666667 = 1.763834208 Reorder the terms: -2.666666667 + x = 1.763834208 Solving -2.666666667 + x = 1.763834208 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.666666667' to each side of the equation. -2.666666667 + 2.666666667 + x = 1.763834208 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + x = 1.763834208 + 2.666666667 x = 1.763834208 + 2.666666667 Combine like terms: 1.763834208 + 2.666666667 = 4.430500875 x = 4.430500875 Simplifying x = 4.430500875Subproblem 2
x + -2.666666667 = -1.763834208 Simplifying x + -2.666666667 = -1.763834208 Reorder the terms: -2.666666667 + x = -1.763834208 Solving -2.666666667 + x = -1.763834208 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.666666667' to each side of the equation. -2.666666667 + 2.666666667 + x = -1.763834208 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + x = -1.763834208 + 2.666666667 x = -1.763834208 + 2.666666667 Combine like terms: -1.763834208 + 2.666666667 = 0.902832459 x = 0.902832459 Simplifying x = 0.902832459Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.430500875, 0.902832459}Solution
x = {4.430500875, 0.902832459}
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