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Simplifying 6a + 4(3a + -7) = a(2a + -5) Reorder the terms: 6a + 4(-7 + 3a) = a(2a + -5) 6a + (-7 * 4 + 3a * 4) = a(2a + -5) 6a + (-28 + 12a) = a(2a + -5) Reorder the terms: -28 + 6a + 12a = a(2a + -5) Combine like terms: 6a + 12a = 18a -28 + 18a = a(2a + -5) Reorder the terms: -28 + 18a = a(-5 + 2a) -28 + 18a = (-5 * a + 2a * a) -28 + 18a = (-5a + 2a2) Solving -28 + 18a = -5a + 2a2 Solving for variable 'a'. Combine like terms: 18a + 5a = 23a -28 + 23a + -2a2 = -5a + 2a2 + 5a + -2a2 Reorder the terms: -28 + 23a + -2a2 = -5a + 5a + 2a2 + -2a2 Combine like terms: -5a + 5a = 0 -28 + 23a + -2a2 = 0 + 2a2 + -2a2 -28 + 23a + -2a2 = 2a2 + -2a2 Combine like terms: 2a2 + -2a2 = 0 -28 + 23a + -2a2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 14 + -11.5a + a2 = 0 Move the constant term to the right: Add '-14' to each side of the equation. 14 + -11.5a + -14 + a2 = 0 + -14 Reorder the terms: 14 + -14 + -11.5a + a2 = 0 + -14 Combine like terms: 14 + -14 = 0 0 + -11.5a + a2 = 0 + -14 -11.5a + a2 = 0 + -14 Combine like terms: 0 + -14 = -14 -11.5a + a2 = -14 The a term is -11.5a. Take half its coefficient (-5.75). Square it (33.0625) and add it to both sides. Add '33.0625' to each side of the equation. -11.5a + 33.0625 + a2 = -14 + 33.0625 Reorder the terms: 33.0625 + -11.5a + a2 = -14 + 33.0625 Combine like terms: -14 + 33.0625 = 19.0625 33.0625 + -11.5a + a2 = 19.0625 Factor a perfect square on the left side: (a + -5.75)(a + -5.75) = 19.0625 Calculate the square root of the right side: 4.366062299 Break this problem into two subproblems by setting (a + -5.75) equal to 4.366062299 and -4.366062299.Subproblem 1
a + -5.75 = 4.366062299 Simplifying a + -5.75 = 4.366062299 Reorder the terms: -5.75 + a = 4.366062299 Solving -5.75 + a = 4.366062299 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '5.75' to each side of the equation. -5.75 + 5.75 + a = 4.366062299 + 5.75 Combine like terms: -5.75 + 5.75 = 0.00 0.00 + a = 4.366062299 + 5.75 a = 4.366062299 + 5.75 Combine like terms: 4.366062299 + 5.75 = 10.116062299 a = 10.116062299 Simplifying a = 10.116062299Subproblem 2
a + -5.75 = -4.366062299 Simplifying a + -5.75 = -4.366062299 Reorder the terms: -5.75 + a = -4.366062299 Solving -5.75 + a = -4.366062299 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '5.75' to each side of the equation. -5.75 + 5.75 + a = -4.366062299 + 5.75 Combine like terms: -5.75 + 5.75 = 0.00 0.00 + a = -4.366062299 + 5.75 a = -4.366062299 + 5.75 Combine like terms: -4.366062299 + 5.75 = 1.383937701 a = 1.383937701 Simplifying a = 1.383937701Solution
The solution to the problem is based on the solutions from the subproblems. a = {10.116062299, 1.383937701}
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