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Simplifying (a + 2)(a + 2) = 60 + -3a Reorder the terms: (2 + a)(a + 2) = 60 + -3a Reorder the terms: (2 + a)(2 + a) = 60 + -3a Multiply (2 + a) * (2 + a) (2(2 + a) + a(2 + a)) = 60 + -3a ((2 * 2 + a * 2) + a(2 + a)) = 60 + -3a ((4 + 2a) + a(2 + a)) = 60 + -3a (4 + 2a + (2 * a + a * a)) = 60 + -3a (4 + 2a + (2a + a2)) = 60 + -3a Combine like terms: 2a + 2a = 4a (4 + 4a + a2) = 60 + -3a Solving 4 + 4a + a2 = 60 + -3a Solving for variable 'a'. Reorder the terms: 4 + -60 + 4a + 3a + a2 = 60 + -3a + -60 + 3a Combine like terms: 4 + -60 = -56 -56 + 4a + 3a + a2 = 60 + -3a + -60 + 3a Combine like terms: 4a + 3a = 7a -56 + 7a + a2 = 60 + -3a + -60 + 3a Reorder the terms: -56 + 7a + a2 = 60 + -60 + -3a + 3a Combine like terms: 60 + -60 = 0 -56 + 7a + a2 = 0 + -3a + 3a -56 + 7a + a2 = -3a + 3a Combine like terms: -3a + 3a = 0 -56 + 7a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '56' to each side of the equation. -56 + 7a + 56 + a2 = 0 + 56 Reorder the terms: -56 + 56 + 7a + a2 = 0 + 56 Combine like terms: -56 + 56 = 0 0 + 7a + a2 = 0 + 56 7a + a2 = 0 + 56 Combine like terms: 0 + 56 = 56 7a + a2 = 56 The a term is 7a. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7a + 12.25 + a2 = 56 + 12.25 Reorder the terms: 12.25 + 7a + a2 = 56 + 12.25 Combine like terms: 56 + 12.25 = 68.25 12.25 + 7a + a2 = 68.25 Factor a perfect square on the left side: (a + 3.5)(a + 3.5) = 68.25 Calculate the square root of the right side: 8.261355821 Break this problem into two subproblems by setting (a + 3.5) equal to 8.261355821 and -8.261355821.Subproblem 1
a + 3.5 = 8.261355821 Simplifying a + 3.5 = 8.261355821 Reorder the terms: 3.5 + a = 8.261355821 Solving 3.5 + a = 8.261355821 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + a = 8.261355821 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + a = 8.261355821 + -3.5 a = 8.261355821 + -3.5 Combine like terms: 8.261355821 + -3.5 = 4.761355821 a = 4.761355821 Simplifying a = 4.761355821Subproblem 2
a + 3.5 = -8.261355821 Simplifying a + 3.5 = -8.261355821 Reorder the terms: 3.5 + a = -8.261355821 Solving 3.5 + a = -8.261355821 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + a = -8.261355821 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + a = -8.261355821 + -3.5 a = -8.261355821 + -3.5 Combine like terms: -8.261355821 + -3.5 = -11.761355821 a = -11.761355821 Simplifying a = -11.761355821Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.761355821, -11.761355821}
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