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Simplifying 12x2 + 74x + -224 = 0 Reorder the terms: -224 + 74x + 12x2 = 0 Solving -224 + 74x + 12x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-112 + 37x + 6x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-112 + 37x + 6x2)' equal to zero and attempt to solve: Simplifying -112 + 37x + 6x2 = 0 Solving -112 + 37x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -18.66666667 + 6.166666667x + x2 = 0 Move the constant term to the right: Add '18.66666667' to each side of the equation. -18.66666667 + 6.166666667x + 18.66666667 + x2 = 0 + 18.66666667 Reorder the terms: -18.66666667 + 18.66666667 + 6.166666667x + x2 = 0 + 18.66666667 Combine like terms: -18.66666667 + 18.66666667 = 0.00000000 0.00000000 + 6.166666667x + x2 = 0 + 18.66666667 6.166666667x + x2 = 0 + 18.66666667 Combine like terms: 0 + 18.66666667 = 18.66666667 6.166666667x + x2 = 18.66666667 The x term is 6.166666667x. Take half its coefficient (3.083333334). Square it (9.506944449) and add it to both sides. Add '9.506944449' to each side of the equation. 6.166666667x + 9.506944449 + x2 = 18.66666667 + 9.506944449 Reorder the terms: 9.506944449 + 6.166666667x + x2 = 18.66666667 + 9.506944449 Combine like terms: 18.66666667 + 9.506944449 = 28.173611119 9.506944449 + 6.166666667x + x2 = 28.173611119 Factor a perfect square on the left side: (x + 3.083333334)(x + 3.083333334) = 28.173611119 Calculate the square root of the right side: 5.307881981 Break this problem into two subproblems by setting (x + 3.083333334) equal to 5.307881981 and -5.307881981.Subproblem 1
x + 3.083333334 = 5.307881981 Simplifying x + 3.083333334 = 5.307881981 Reorder the terms: 3.083333334 + x = 5.307881981 Solving 3.083333334 + x = 5.307881981 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.083333334' to each side of the equation. 3.083333334 + -3.083333334 + x = 5.307881981 + -3.083333334 Combine like terms: 3.083333334 + -3.083333334 = 0.000000000 0.000000000 + x = 5.307881981 + -3.083333334 x = 5.307881981 + -3.083333334 Combine like terms: 5.307881981 + -3.083333334 = 2.224548647 x = 2.224548647 Simplifying x = 2.224548647Subproblem 2
x + 3.083333334 = -5.307881981 Simplifying x + 3.083333334 = -5.307881981 Reorder the terms: 3.083333334 + x = -5.307881981 Solving 3.083333334 + x = -5.307881981 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.083333334' to each side of the equation. 3.083333334 + -3.083333334 + x = -5.307881981 + -3.083333334 Combine like terms: 3.083333334 + -3.083333334 = 0.000000000 0.000000000 + x = -5.307881981 + -3.083333334 x = -5.307881981 + -3.083333334 Combine like terms: -5.307881981 + -3.083333334 = -8.391215315 x = -8.391215315 Simplifying x = -8.391215315Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.224548647, -8.391215315}Solution
x = {2.224548647, -8.391215315}
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