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Simplifying 54x2 + 4x + -24 = 0 Reorder the terms: -24 + 4x + 54x2 = 0 Solving -24 + 4x + 54x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-12 + 2x + 27x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-12 + 2x + 27x2)' equal to zero and attempt to solve: Simplifying -12 + 2x + 27x2 = 0 Solving -12 + 2x + 27x2 = 0 Begin completing the square. Divide all terms by 27 the coefficient of the squared term: Divide each side by '27'. -0.4444444444 + 0.07407407407x + x2 = 0 Move the constant term to the right: Add '0.4444444444' to each side of the equation. -0.4444444444 + 0.07407407407x + 0.4444444444 + x2 = 0 + 0.4444444444 Reorder the terms: -0.4444444444 + 0.4444444444 + 0.07407407407x + x2 = 0 + 0.4444444444 Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000 0.0000000000 + 0.07407407407x + x2 = 0 + 0.4444444444 0.07407407407x + x2 = 0 + 0.4444444444 Combine like terms: 0 + 0.4444444444 = 0.4444444444 0.07407407407x + x2 = 0.4444444444 The x term is 0.07407407407x. Take half its coefficient (0.03703703704). Square it (0.001371742113) and add it to both sides. Add '0.001371742113' to each side of the equation. 0.07407407407x + 0.001371742113 + x2 = 0.4444444444 + 0.001371742113 Reorder the terms: 0.001371742113 + 0.07407407407x + x2 = 0.4444444444 + 0.001371742113 Combine like terms: 0.4444444444 + 0.001371742113 = 0.445816186513 0.001371742113 + 0.07407407407x + x2 = 0.445816186513 Factor a perfect square on the left side: (x + 0.03703703704)(x + 0.03703703704) = 0.445816186513 Calculate the square root of the right side: 0.667694681 Break this problem into two subproblems by setting (x + 0.03703703704) equal to 0.667694681 and -0.667694681.Subproblem 1
x + 0.03703703704 = 0.667694681 Simplifying x + 0.03703703704 = 0.667694681 Reorder the terms: 0.03703703704 + x = 0.667694681 Solving 0.03703703704 + x = 0.667694681 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.03703703704' to each side of the equation. 0.03703703704 + -0.03703703704 + x = 0.667694681 + -0.03703703704 Combine like terms: 0.03703703704 + -0.03703703704 = 0.00000000000 0.00000000000 + x = 0.667694681 + -0.03703703704 x = 0.667694681 + -0.03703703704 Combine like terms: 0.667694681 + -0.03703703704 = 0.63065764396 x = 0.63065764396 Simplifying x = 0.63065764396Subproblem 2
x + 0.03703703704 = -0.667694681 Simplifying x + 0.03703703704 = -0.667694681 Reorder the terms: 0.03703703704 + x = -0.667694681 Solving 0.03703703704 + x = -0.667694681 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.03703703704' to each side of the equation. 0.03703703704 + -0.03703703704 + x = -0.667694681 + -0.03703703704 Combine like terms: 0.03703703704 + -0.03703703704 = 0.00000000000 0.00000000000 + x = -0.667694681 + -0.03703703704 x = -0.667694681 + -0.03703703704 Combine like terms: -0.667694681 + -0.03703703704 = -0.70473171804 x = -0.70473171804 Simplifying x = -0.70473171804Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.63065764396, -0.70473171804}Solution
x = {0.63065764396, -0.70473171804}
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