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Simplifying 12x2 + 24x = 114 Reorder the terms: 24x + 12x2 = 114 Solving 24x + 12x2 = 114 Solving for variable 'x'. Reorder the terms: -114 + 24x + 12x2 = 114 + -114 Combine like terms: 114 + -114 = 0 -114 + 24x + 12x2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-19 + 4x + 2x2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-19 + 4x + 2x2)' equal to zero and attempt to solve: Simplifying -19 + 4x + 2x2 = 0 Solving -19 + 4x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -9.5 + 2x + x2 = 0 Move the constant term to the right: Add '9.5' to each side of the equation. -9.5 + 2x + 9.5 + x2 = 0 + 9.5 Reorder the terms: -9.5 + 9.5 + 2x + x2 = 0 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + 2x + x2 = 0 + 9.5 2x + x2 = 0 + 9.5 Combine like terms: 0 + 9.5 = 9.5 2x + x2 = 9.5 The x term is 2x. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2x + 1 + x2 = 9.5 + 1 Reorder the terms: 1 + 2x + x2 = 9.5 + 1 Combine like terms: 9.5 + 1 = 10.5 1 + 2x + x2 = 10.5 Factor a perfect square on the left side: (x + 1)(x + 1) = 10.5 Calculate the square root of the right side: 3.240370349 Break this problem into two subproblems by setting (x + 1) equal to 3.240370349 and -3.240370349.Subproblem 1
x + 1 = 3.240370349 Simplifying x + 1 = 3.240370349 Reorder the terms: 1 + x = 3.240370349 Solving 1 + x = 3.240370349 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 3.240370349 + -1 Combine like terms: 1 + -1 = 0 0 + x = 3.240370349 + -1 x = 3.240370349 + -1 Combine like terms: 3.240370349 + -1 = 2.240370349 x = 2.240370349 Simplifying x = 2.240370349Subproblem 2
x + 1 = -3.240370349 Simplifying x + 1 = -3.240370349 Reorder the terms: 1 + x = -3.240370349 Solving 1 + x = -3.240370349 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -3.240370349 + -1 Combine like terms: 1 + -1 = 0 0 + x = -3.240370349 + -1 x = -3.240370349 + -1 Combine like terms: -3.240370349 + -1 = -4.240370349 x = -4.240370349 Simplifying x = -4.240370349Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.240370349, -4.240370349}Solution
x = {2.240370349, -4.240370349}
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