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Simplifying 12x2 + 32x + -69 = 0 Reorder the terms: -69 + 32x + 12x2 = 0 Solving -69 + 32x + 12x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -5.75 + 2.666666667x + x2 = 0 Move the constant term to the right: Add '5.75' to each side of the equation. -5.75 + 2.666666667x + 5.75 + x2 = 0 + 5.75 Reorder the terms: -5.75 + 5.75 + 2.666666667x + x2 = 0 + 5.75 Combine like terms: -5.75 + 5.75 = 0.00 0.00 + 2.666666667x + x2 = 0 + 5.75 2.666666667x + x2 = 0 + 5.75 Combine like terms: 0 + 5.75 = 5.75 2.666666667x + x2 = 5.75 The x term is 2.666666667x. Take half its coefficient (1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. 2.666666667x + 1.777777780 + x2 = 5.75 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667x + x2 = 5.75 + 1.777777780 Combine like terms: 5.75 + 1.777777780 = 7.52777778 1.777777780 + 2.666666667x + x2 = 7.52777778 Factor a perfect square on the left side: (x + 1.333333334)(x + 1.333333334) = 7.52777778 Calculate the square root of the right side: 2.743679606 Break this problem into two subproblems by setting (x + 1.333333334) equal to 2.743679606 and -2.743679606.Subproblem 1
x + 1.333333334 = 2.743679606 Simplifying x + 1.333333334 = 2.743679606 Reorder the terms: 1.333333334 + x = 2.743679606 Solving 1.333333334 + x = 2.743679606 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = 2.743679606 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 2.743679606 + -1.333333334 x = 2.743679606 + -1.333333334 Combine like terms: 2.743679606 + -1.333333334 = 1.410346272 x = 1.410346272 Simplifying x = 1.410346272Subproblem 2
x + 1.333333334 = -2.743679606 Simplifying x + 1.333333334 = -2.743679606 Reorder the terms: 1.333333334 + x = -2.743679606 Solving 1.333333334 + x = -2.743679606 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = -2.743679606 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -2.743679606 + -1.333333334 x = -2.743679606 + -1.333333334 Combine like terms: -2.743679606 + -1.333333334 = -4.07701294 x = -4.07701294 Simplifying x = -4.07701294Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.410346272, -4.07701294}
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