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Simplifying 3a(a + 6) + -2(a + -6) = 6 Reorder the terms: 3a(6 + a) + -2(a + -6) = 6 (6 * 3a + a * 3a) + -2(a + -6) = 6 (18a + 3a2) + -2(a + -6) = 6 Reorder the terms: 18a + 3a2 + -2(-6 + a) = 6 18a + 3a2 + (-6 * -2 + a * -2) = 6 18a + 3a2 + (12 + -2a) = 6 Reorder the terms: 12 + 18a + -2a + 3a2 = 6 Combine like terms: 18a + -2a = 16a 12 + 16a + 3a2 = 6 Solving 12 + 16a + 3a2 = 6 Solving for variable 'a'. Reorder the terms: 12 + -6 + 16a + 3a2 = 6 + -6 Combine like terms: 12 + -6 = 6 6 + 16a + 3a2 = 6 + -6 Combine like terms: 6 + -6 = 0 6 + 16a + 3a2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 2 + 5.333333333a + a2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + 5.333333333a + -2 + a2 = 0 + -2 Reorder the terms: 2 + -2 + 5.333333333a + a2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + 5.333333333a + a2 = 0 + -2 5.333333333a + a2 = 0 + -2 Combine like terms: 0 + -2 = -2 5.333333333a + a2 = -2 The a term is 5.333333333a. Take half its coefficient (2.666666667). Square it (7.111111113) and add it to both sides. Add '7.111111113' to each side of the equation. 5.333333333a + 7.111111113 + a2 = -2 + 7.111111113 Reorder the terms: 7.111111113 + 5.333333333a + a2 = -2 + 7.111111113 Combine like terms: -2 + 7.111111113 = 5.111111113 7.111111113 + 5.333333333a + a2 = 5.111111113 Factor a perfect square on the left side: (a + 2.666666667)(a + 2.666666667) = 5.111111113 Calculate the square root of the right side: 2.260776661 Break this problem into two subproblems by setting (a + 2.666666667) equal to 2.260776661 and -2.260776661.Subproblem 1
a + 2.666666667 = 2.260776661 Simplifying a + 2.666666667 = 2.260776661 Reorder the terms: 2.666666667 + a = 2.260776661 Solving 2.666666667 + a = 2.260776661 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + a = 2.260776661 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + a = 2.260776661 + -2.666666667 a = 2.260776661 + -2.666666667 Combine like terms: 2.260776661 + -2.666666667 = -0.405890006 a = -0.405890006 Simplifying a = -0.405890006Subproblem 2
a + 2.666666667 = -2.260776661 Simplifying a + 2.666666667 = -2.260776661 Reorder the terms: 2.666666667 + a = -2.260776661 Solving 2.666666667 + a = -2.260776661 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + a = -2.260776661 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + a = -2.260776661 + -2.666666667 a = -2.260776661 + -2.666666667 Combine like terms: -2.260776661 + -2.666666667 = -4.927443328 a = -4.927443328 Simplifying a = -4.927443328Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.405890006, -4.927443328}
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