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Simplifying 18x2 + 12x = 22 Reorder the terms: 12x + 18x2 = 22 Solving 12x + 18x2 = 22 Solving for variable 'x'. Reorder the terms: -22 + 12x + 18x2 = 22 + -22 Combine like terms: 22 + -22 = 0 -22 + 12x + 18x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-11 + 6x + 9x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-11 + 6x + 9x2)' equal to zero and attempt to solve: Simplifying -11 + 6x + 9x2 = 0 Solving -11 + 6x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1.222222222 + 0.6666666667x + x2 = 0 Move the constant term to the right: Add '1.222222222' to each side of the equation. -1.222222222 + 0.6666666667x + 1.222222222 + x2 = 0 + 1.222222222 Reorder the terms: -1.222222222 + 1.222222222 + 0.6666666667x + x2 = 0 + 1.222222222 Combine like terms: -1.222222222 + 1.222222222 = 0.000000000 0.000000000 + 0.6666666667x + x2 = 0 + 1.222222222 0.6666666667x + x2 = 0 + 1.222222222 Combine like terms: 0 + 1.222222222 = 1.222222222 0.6666666667x + x2 = 1.222222222 The x term is 0.6666666667x. Take half its coefficient (0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. 0.6666666667x + 0.1111111112 + x2 = 1.222222222 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667x + x2 = 1.222222222 + 0.1111111112 Combine like terms: 1.222222222 + 0.1111111112 = 1.3333333332 0.1111111112 + 0.6666666667x + x2 = 1.3333333332 Factor a perfect square on the left side: (x + 0.3333333334)(x + 0.3333333334) = 1.3333333332 Calculate the square root of the right side: 1.154700538 Break this problem into two subproblems by setting (x + 0.3333333334) equal to 1.154700538 and -1.154700538.Subproblem 1
x + 0.3333333334 = 1.154700538 Simplifying x + 0.3333333334 = 1.154700538 Reorder the terms: 0.3333333334 + x = 1.154700538 Solving 0.3333333334 + x = 1.154700538 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = 1.154700538 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = 1.154700538 + -0.3333333334 x = 1.154700538 + -0.3333333334 Combine like terms: 1.154700538 + -0.3333333334 = 0.8213672046 x = 0.8213672046 Simplifying x = 0.8213672046Subproblem 2
x + 0.3333333334 = -1.154700538 Simplifying x + 0.3333333334 = -1.154700538 Reorder the terms: 0.3333333334 + x = -1.154700538 Solving 0.3333333334 + x = -1.154700538 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = -1.154700538 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = -1.154700538 + -0.3333333334 x = -1.154700538 + -0.3333333334 Combine like terms: -1.154700538 + -0.3333333334 = -1.4880338714 x = -1.4880338714 Simplifying x = -1.4880338714Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.8213672046, -1.4880338714}Solution
x = {0.8213672046, -1.4880338714}
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