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Simplifying (2a + 3)(2a + 3) = 5 Reorder the terms: (3 + 2a)(2a + 3) = 5 Reorder the terms: (3 + 2a)(3 + 2a) = 5 Multiply (3 + 2a) * (3 + 2a) (3(3 + 2a) + 2a * (3 + 2a)) = 5 ((3 * 3 + 2a * 3) + 2a * (3 + 2a)) = 5 ((9 + 6a) + 2a * (3 + 2a)) = 5 (9 + 6a + (3 * 2a + 2a * 2a)) = 5 (9 + 6a + (6a + 4a2)) = 5 Combine like terms: 6a + 6a = 12a (9 + 12a + 4a2) = 5 Solving 9 + 12a + 4a2 = 5 Solving for variable 'a'. Reorder the terms: 9 + -5 + 12a + 4a2 = 5 + -5 Combine like terms: 9 + -5 = 4 4 + 12a + 4a2 = 5 + -5 Combine like terms: 5 + -5 = 0 4 + 12a + 4a2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(1 + 3a + a2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(1 + 3a + a2)' equal to zero and attempt to solve: Simplifying 1 + 3a + a2 = 0 Solving 1 + 3a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + 3a + -1 + a2 = 0 + -1 Reorder the terms: 1 + -1 + 3a + a2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 3a + a2 = 0 + -1 3a + a2 = 0 + -1 Combine like terms: 0 + -1 = -1 3a + a2 = -1 The a term is 3a. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3a + 2.25 + a2 = -1 + 2.25 Reorder the terms: 2.25 + 3a + a2 = -1 + 2.25 Combine like terms: -1 + 2.25 = 1.25 2.25 + 3a + a2 = 1.25 Factor a perfect square on the left side: (a + 1.5)(a + 1.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (a + 1.5) equal to 1.118033989 and -1.118033989.Subproblem 1
a + 1.5 = 1.118033989 Simplifying a + 1.5 = 1.118033989 Reorder the terms: 1.5 + a = 1.118033989 Solving 1.5 + a = 1.118033989 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = 1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = 1.118033989 + -1.5 a = 1.118033989 + -1.5 Combine like terms: 1.118033989 + -1.5 = -0.381966011 a = -0.381966011 Simplifying a = -0.381966011Subproblem 2
a + 1.5 = -1.118033989 Simplifying a + 1.5 = -1.118033989 Reorder the terms: 1.5 + a = -1.118033989 Solving 1.5 + a = -1.118033989 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = -1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = -1.118033989 + -1.5 a = -1.118033989 + -1.5 Combine like terms: -1.118033989 + -1.5 = -2.618033989 a = -2.618033989 Simplifying a = -2.618033989Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.381966011, -2.618033989}Solution
a = {-0.381966011, -2.618033989}
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