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Simplifying 5a(2a + 12) = 180 Reorder the terms: 5a(12 + 2a) = 180 (12 * 5a + 2a * 5a) = 180 (60a + 10a2) = 180 Solving 60a + 10a2 = 180 Solving for variable 'a'. Reorder the terms: -180 + 60a + 10a2 = 180 + -180 Combine like terms: 180 + -180 = 0 -180 + 60a + 10a2 = 0 Factor out the Greatest Common Factor (GCF), '10'. 10(-18 + 6a + a2) = 0 Ignore the factor 10.Subproblem 1
Set the factor '(-18 + 6a + a2)' equal to zero and attempt to solve: Simplifying -18 + 6a + a2 = 0 Solving -18 + 6a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '18' to each side of the equation. -18 + 6a + 18 + a2 = 0 + 18 Reorder the terms: -18 + 18 + 6a + a2 = 0 + 18 Combine like terms: -18 + 18 = 0 0 + 6a + a2 = 0 + 18 6a + a2 = 0 + 18 Combine like terms: 0 + 18 = 18 6a + a2 = 18 The a term is 6a. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6a + 9 + a2 = 18 + 9 Reorder the terms: 9 + 6a + a2 = 18 + 9 Combine like terms: 18 + 9 = 27 9 + 6a + a2 = 27 Factor a perfect square on the left side: (a + 3)(a + 3) = 27 Calculate the square root of the right side: 5.196152423 Break this problem into two subproblems by setting (a + 3) equal to 5.196152423 and -5.196152423.Subproblem 1
a + 3 = 5.196152423 Simplifying a + 3 = 5.196152423 Reorder the terms: 3 + a = 5.196152423 Solving 3 + a = 5.196152423 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + a = 5.196152423 + -3 Combine like terms: 3 + -3 = 0 0 + a = 5.196152423 + -3 a = 5.196152423 + -3 Combine like terms: 5.196152423 + -3 = 2.196152423 a = 2.196152423 Simplifying a = 2.196152423Subproblem 2
a + 3 = -5.196152423 Simplifying a + 3 = -5.196152423 Reorder the terms: 3 + a = -5.196152423 Solving 3 + a = -5.196152423 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + a = -5.196152423 + -3 Combine like terms: 3 + -3 = 0 0 + a = -5.196152423 + -3 a = -5.196152423 + -3 Combine like terms: -5.196152423 + -3 = -8.196152423 a = -8.196152423 Simplifying a = -8.196152423Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.196152423, -8.196152423}Solution
a = {2.196152423, -8.196152423}
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