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Simplifying (a + -3)(a + 8) = 24 Reorder the terms: (-3 + a)(a + 8) = 24 Reorder the terms: (-3 + a)(8 + a) = 24 Multiply (-3 + a) * (8 + a) (-3(8 + a) + a(8 + a)) = 24 ((8 * -3 + a * -3) + a(8 + a)) = 24 ((-24 + -3a) + a(8 + a)) = 24 (-24 + -3a + (8 * a + a * a)) = 24 (-24 + -3a + (8a + a2)) = 24 Combine like terms: -3a + 8a = 5a (-24 + 5a + a2) = 24 Solving -24 + 5a + a2 = 24 Solving for variable 'a'. Reorder the terms: -24 + -24 + 5a + a2 = 24 + -24 Combine like terms: -24 + -24 = -48 -48 + 5a + a2 = 24 + -24 Combine like terms: 24 + -24 = 0 -48 + 5a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '48' to each side of the equation. -48 + 5a + 48 + a2 = 0 + 48 Reorder the terms: -48 + 48 + 5a + a2 = 0 + 48 Combine like terms: -48 + 48 = 0 0 + 5a + a2 = 0 + 48 5a + a2 = 0 + 48 Combine like terms: 0 + 48 = 48 5a + a2 = 48 The a term is 5a. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5a + 6.25 + a2 = 48 + 6.25 Reorder the terms: 6.25 + 5a + a2 = 48 + 6.25 Combine like terms: 48 + 6.25 = 54.25 6.25 + 5a + a2 = 54.25 Factor a perfect square on the left side: (a + 2.5)(a + 2.5) = 54.25 Calculate the square root of the right side: 7.365459931 Break this problem into two subproblems by setting (a + 2.5) equal to 7.365459931 and -7.365459931.Subproblem 1
a + 2.5 = 7.365459931 Simplifying a + 2.5 = 7.365459931 Reorder the terms: 2.5 + a = 7.365459931 Solving 2.5 + a = 7.365459931 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + a = 7.365459931 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + a = 7.365459931 + -2.5 a = 7.365459931 + -2.5 Combine like terms: 7.365459931 + -2.5 = 4.865459931 a = 4.865459931 Simplifying a = 4.865459931Subproblem 2
a + 2.5 = -7.365459931 Simplifying a + 2.5 = -7.365459931 Reorder the terms: 2.5 + a = -7.365459931 Solving 2.5 + a = -7.365459931 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + a = -7.365459931 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + a = -7.365459931 + -2.5 a = -7.365459931 + -2.5 Combine like terms: -7.365459931 + -2.5 = -9.865459931 a = -9.865459931 Simplifying a = -9.865459931Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.865459931, -9.865459931}
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