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Simplifying 4(7a + -9) + a(6a + 3) = 0 Reorder the terms: 4(-9 + 7a) + a(6a + 3) = 0 (-9 * 4 + 7a * 4) + a(6a + 3) = 0 (-36 + 28a) + a(6a + 3) = 0 Reorder the terms: -36 + 28a + a(3 + 6a) = 0 -36 + 28a + (3 * a + 6a * a) = 0 -36 + 28a + (3a + 6a2) = 0 Combine like terms: 28a + 3a = 31a -36 + 31a + 6a2 = 0 Solving -36 + 31a + 6a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -6 + 5.166666667a + a2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + 5.166666667a + 6 + a2 = 0 + 6 Reorder the terms: -6 + 6 + 5.166666667a + a2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + 5.166666667a + a2 = 0 + 6 5.166666667a + a2 = 0 + 6 Combine like terms: 0 + 6 = 6 5.166666667a + a2 = 6 The a term is 5.166666667a. Take half its coefficient (2.583333334). Square it (6.673611115) and add it to both sides. Add '6.673611115' to each side of the equation. 5.166666667a + 6.673611115 + a2 = 6 + 6.673611115 Reorder the terms: 6.673611115 + 5.166666667a + a2 = 6 + 6.673611115 Combine like terms: 6 + 6.673611115 = 12.673611115 6.673611115 + 5.166666667a + a2 = 12.673611115 Factor a perfect square on the left side: (a + 2.583333334)(a + 2.583333334) = 12.673611115 Calculate the square root of the right side: 3.560001561 Break this problem into two subproblems by setting (a + 2.583333334) equal to 3.560001561 and -3.560001561.Subproblem 1
a + 2.583333334 = 3.560001561 Simplifying a + 2.583333334 = 3.560001561 Reorder the terms: 2.583333334 + a = 3.560001561 Solving 2.583333334 + a = 3.560001561 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.583333334' to each side of the equation. 2.583333334 + -2.583333334 + a = 3.560001561 + -2.583333334 Combine like terms: 2.583333334 + -2.583333334 = 0.000000000 0.000000000 + a = 3.560001561 + -2.583333334 a = 3.560001561 + -2.583333334 Combine like terms: 3.560001561 + -2.583333334 = 0.976668227 a = 0.976668227 Simplifying a = 0.976668227Subproblem 2
a + 2.583333334 = -3.560001561 Simplifying a + 2.583333334 = -3.560001561 Reorder the terms: 2.583333334 + a = -3.560001561 Solving 2.583333334 + a = -3.560001561 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.583333334' to each side of the equation. 2.583333334 + -2.583333334 + a = -3.560001561 + -2.583333334 Combine like terms: 2.583333334 + -2.583333334 = 0.000000000 0.000000000 + a = -3.560001561 + -2.583333334 a = -3.560001561 + -2.583333334 Combine like terms: -3.560001561 + -2.583333334 = -6.143334895 a = -6.143334895 Simplifying a = -6.143334895Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.976668227, -6.143334895}
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