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Simplifying 12x(5 + 6x) = 54 (5 * 12x + 6x * 12x) = 54 (60x + 72x2) = 54 Solving 60x + 72x2 = 54 Solving for variable 'x'. Reorder the terms: -54 + 60x + 72x2 = 54 + -54 Combine like terms: 54 + -54 = 0 -54 + 60x + 72x2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-9 + 10x + 12x2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-9 + 10x + 12x2)' equal to zero and attempt to solve: Simplifying -9 + 10x + 12x2 = 0 Solving -9 + 10x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.75 + 0.8333333333x + x2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 0.8333333333x + 0.75 + x2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 0.8333333333x + x2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 0.8333333333x + x2 = 0 + 0.75 0.8333333333x + x2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 0.8333333333x + x2 = 0.75 The x term is 0.8333333333x. Take half its coefficient (0.4166666667). Square it (0.1736111111) and add it to both sides. Add '0.1736111111' to each side of the equation. 0.8333333333x + 0.1736111111 + x2 = 0.75 + 0.1736111111 Reorder the terms: 0.1736111111 + 0.8333333333x + x2 = 0.75 + 0.1736111111 Combine like terms: 0.75 + 0.1736111111 = 0.9236111111 0.1736111111 + 0.8333333333x + x2 = 0.9236111111 Factor a perfect square on the left side: (x + 0.4166666667)(x + 0.4166666667) = 0.9236111111 Calculate the square root of the right side: 0.961046883 Break this problem into two subproblems by setting (x + 0.4166666667) equal to 0.961046883 and -0.961046883.Subproblem 1
x + 0.4166666667 = 0.961046883 Simplifying x + 0.4166666667 = 0.961046883 Reorder the terms: 0.4166666667 + x = 0.961046883 Solving 0.4166666667 + x = 0.961046883 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + x = 0.961046883 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = 0.961046883 + -0.4166666667 x = 0.961046883 + -0.4166666667 Combine like terms: 0.961046883 + -0.4166666667 = 0.5443802163 x = 0.5443802163 Simplifying x = 0.5443802163Subproblem 2
x + 0.4166666667 = -0.961046883 Simplifying x + 0.4166666667 = -0.961046883 Reorder the terms: 0.4166666667 + x = -0.961046883 Solving 0.4166666667 + x = -0.961046883 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + x = -0.961046883 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = -0.961046883 + -0.4166666667 x = -0.961046883 + -0.4166666667 Combine like terms: -0.961046883 + -0.4166666667 = -1.3777135497 x = -1.3777135497 Simplifying x = -1.3777135497Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.5443802163, -1.3777135497}Solution
x = {0.5443802163, -1.3777135497}
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