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Simplifying 5(6a + -7) + -6a = 5(a + 8) * 4a Reorder the terms: 5(-7 + 6a) + -6a = 5(a + 8) * 4a (-7 * 5 + 6a * 5) + -6a = 5(a + 8) * 4a (-35 + 30a) + -6a = 5(a + 8) * 4a Combine like terms: 30a + -6a = 24a -35 + 24a = 5(a + 8) * 4a Reorder the terms: -35 + 24a = 5(8 + a) * 4a Reorder the terms for easier multiplication: -35 + 24a = 5 * 4a(8 + a) Multiply 5 * 4 -35 + 24a = 20a(8 + a) -35 + 24a = (8 * 20a + a * 20a) -35 + 24a = (160a + 20a2) Solving -35 + 24a = 160a + 20a2 Solving for variable 'a'. Combine like terms: 24a + -160a = -136a -35 + -136a + -20a2 = 160a + 20a2 + -160a + -20a2 Reorder the terms: -35 + -136a + -20a2 = 160a + -160a + 20a2 + -20a2 Combine like terms: 160a + -160a = 0 -35 + -136a + -20a2 = 0 + 20a2 + -20a2 -35 + -136a + -20a2 = 20a2 + -20a2 Combine like terms: 20a2 + -20a2 = 0 -35 + -136a + -20a2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(35 + 136a + 20a2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(35 + 136a + 20a2)' equal to zero and attempt to solve: Simplifying 35 + 136a + 20a2 = 0 Solving 35 + 136a + 20a2 = 0 Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. 1.75 + 6.8a + a2 = 0.0 Move the constant term to the right: Add '-1.75' to each side of the equation. 1.75 + 6.8a + -1.75 + a2 = 0.0 + -1.75 Reorder the terms: 1.75 + -1.75 + 6.8a + a2 = 0.0 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + 6.8a + a2 = 0.0 + -1.75 6.8a + a2 = 0.0 + -1.75 Combine like terms: 0.0 + -1.75 = -1.75 6.8a + a2 = -1.75 The a term is 6.8a. Take half its coefficient (3.4). Square it (11.56) and add it to both sides. Add '11.56' to each side of the equation. 6.8a + 11.56 + a2 = -1.75 + 11.56 Reorder the terms: 11.56 + 6.8a + a2 = -1.75 + 11.56 Combine like terms: -1.75 + 11.56 = 9.81 11.56 + 6.8a + a2 = 9.81 Factor a perfect square on the left side: (a + 3.4)(a + 3.4) = 9.81 Calculate the square root of the right side: 3.132091953 Break this problem into two subproblems by setting (a + 3.4) equal to 3.132091953 and -3.132091953.Subproblem 1
a + 3.4 = 3.132091953 Simplifying a + 3.4 = 3.132091953 Reorder the terms: 3.4 + a = 3.132091953 Solving 3.4 + a = 3.132091953 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3.4' to each side of the equation. 3.4 + -3.4 + a = 3.132091953 + -3.4 Combine like terms: 3.4 + -3.4 = 0.0 0.0 + a = 3.132091953 + -3.4 a = 3.132091953 + -3.4 Combine like terms: 3.132091953 + -3.4 = -0.267908047 a = -0.267908047 Simplifying a = -0.267908047Subproblem 2
a + 3.4 = -3.132091953 Simplifying a + 3.4 = -3.132091953 Reorder the terms: 3.4 + a = -3.132091953 Solving 3.4 + a = -3.132091953 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3.4' to each side of the equation. 3.4 + -3.4 + a = -3.132091953 + -3.4 Combine like terms: 3.4 + -3.4 = 0.0 0.0 + a = -3.132091953 + -3.4 a = -3.132091953 + -3.4 Combine like terms: -3.132091953 + -3.4 = -6.532091953 a = -6.532091953 Simplifying a = -6.532091953Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.267908047, -6.532091953}Solution
a = {-0.267908047, -6.532091953}
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