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Simplifying (4a + -3)(5a + 5) = 8 Reorder the terms: (-3 + 4a)(5a + 5) = 8 Reorder the terms: (-3 + 4a)(5 + 5a) = 8 Multiply (-3 + 4a) * (5 + 5a) (-3(5 + 5a) + 4a * (5 + 5a)) = 8 ((5 * -3 + 5a * -3) + 4a * (5 + 5a)) = 8 ((-15 + -15a) + 4a * (5 + 5a)) = 8 (-15 + -15a + (5 * 4a + 5a * 4a)) = 8 (-15 + -15a + (20a + 20a2)) = 8 Combine like terms: -15a + 20a = 5a (-15 + 5a + 20a2) = 8 Solving -15 + 5a + 20a2 = 8 Solving for variable 'a'. Reorder the terms: -15 + -8 + 5a + 20a2 = 8 + -8 Combine like terms: -15 + -8 = -23 -23 + 5a + 20a2 = 8 + -8 Combine like terms: 8 + -8 = 0 -23 + 5a + 20a2 = 0 Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. -1.15 + 0.25a + a2 = 0.0 Move the constant term to the right: Add '1.15' to each side of the equation. -1.15 + 0.25a + 1.15 + a2 = 0.0 + 1.15 Reorder the terms: -1.15 + 1.15 + 0.25a + a2 = 0.0 + 1.15 Combine like terms: -1.15 + 1.15 = 0.00 0.00 + 0.25a + a2 = 0.0 + 1.15 0.25a + a2 = 0.0 + 1.15 Combine like terms: 0.0 + 1.15 = 1.15 0.25a + a2 = 1.15 The a term is 0.25a. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25a + 0.015625 + a2 = 1.15 + 0.015625 Reorder the terms: 0.015625 + 0.25a + a2 = 1.15 + 0.015625 Combine like terms: 1.15 + 0.015625 = 1.165625 0.015625 + 0.25a + a2 = 1.165625 Factor a perfect square on the left side: (a + 0.125)(a + 0.125) = 1.165625 Calculate the square root of the right side: 1.079641144 Break this problem into two subproblems by setting (a + 0.125) equal to 1.079641144 and -1.079641144.Subproblem 1
a + 0.125 = 1.079641144 Simplifying a + 0.125 = 1.079641144 Reorder the terms: 0.125 + a = 1.079641144 Solving 0.125 + a = 1.079641144 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + a = 1.079641144 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + a = 1.079641144 + -0.125 a = 1.079641144 + -0.125 Combine like terms: 1.079641144 + -0.125 = 0.954641144 a = 0.954641144 Simplifying a = 0.954641144Subproblem 2
a + 0.125 = -1.079641144 Simplifying a + 0.125 = -1.079641144 Reorder the terms: 0.125 + a = -1.079641144 Solving 0.125 + a = -1.079641144 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + a = -1.079641144 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + a = -1.079641144 + -0.125 a = -1.079641144 + -0.125 Combine like terms: -1.079641144 + -0.125 = -1.204641144 a = -1.204641144 Simplifying a = -1.204641144Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.954641144, -1.204641144}
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