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Simplifying 5a * 5(a + 3) + (4 + 8a) * 6 = 0 Reorder the terms: 5a * 5(3 + a) + (4 + 8a) * 6 = 0 Reorder the terms for easier multiplication: 5 * 5a(3 + a) + (4 + 8a) * 6 = 0 Multiply 5 * 5 25a(3 + a) + (4 + 8a) * 6 = 0 (3 * 25a + a * 25a) + (4 + 8a) * 6 = 0 (75a + 25a2) + (4 + 8a) * 6 = 0 Reorder the terms for easier multiplication: 75a + 25a2 + 6(4 + 8a) = 0 75a + 25a2 + (4 * 6 + 8a * 6) = 0 75a + 25a2 + (24 + 48a) = 0 Reorder the terms: 24 + 75a + 48a + 25a2 = 0 Combine like terms: 75a + 48a = 123a 24 + 123a + 25a2 = 0 Solving 24 + 123a + 25a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 25 the coefficient of the squared term: Divide each side by '25'. 0.96 + 4.92a + a2 = 0 Move the constant term to the right: Add '-0.96' to each side of the equation. 0.96 + 4.92a + -0.96 + a2 = 0 + -0.96 Reorder the terms: 0.96 + -0.96 + 4.92a + a2 = 0 + -0.96 Combine like terms: 0.96 + -0.96 = 0.00 0.00 + 4.92a + a2 = 0 + -0.96 4.92a + a2 = 0 + -0.96 Combine like terms: 0 + -0.96 = -0.96 4.92a + a2 = -0.96 The a term is 4.92a. Take half its coefficient (2.46). Square it (6.0516) and add it to both sides. Add '6.0516' to each side of the equation. 4.92a + 6.0516 + a2 = -0.96 + 6.0516 Reorder the terms: 6.0516 + 4.92a + a2 = -0.96 + 6.0516 Combine like terms: -0.96 + 6.0516 = 5.0916 6.0516 + 4.92a + a2 = 5.0916 Factor a perfect square on the left side: (a + 2.46)(a + 2.46) = 5.0916 Calculate the square root of the right side: 2.2564574 Break this problem into two subproblems by setting (a + 2.46) equal to 2.2564574 and -2.2564574.Subproblem 1
a + 2.46 = 2.2564574 Simplifying a + 2.46 = 2.2564574 Reorder the terms: 2.46 + a = 2.2564574 Solving 2.46 + a = 2.2564574 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.46' to each side of the equation. 2.46 + -2.46 + a = 2.2564574 + -2.46 Combine like terms: 2.46 + -2.46 = 0.00 0.00 + a = 2.2564574 + -2.46 a = 2.2564574 + -2.46 Combine like terms: 2.2564574 + -2.46 = -0.2035426 a = -0.2035426 Simplifying a = -0.2035426Subproblem 2
a + 2.46 = -2.2564574 Simplifying a + 2.46 = -2.2564574 Reorder the terms: 2.46 + a = -2.2564574 Solving 2.46 + a = -2.2564574 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.46' to each side of the equation. 2.46 + -2.46 + a = -2.2564574 + -2.46 Combine like terms: 2.46 + -2.46 = 0.00 0.00 + a = -2.2564574 + -2.46 a = -2.2564574 + -2.46 Combine like terms: -2.2564574 + -2.46 = -4.7164574 a = -4.7164574 Simplifying a = -4.7164574Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.2035426, -4.7164574}
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