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Simplifying 2a(a + 24) = 30 Reorder the terms: 2a(24 + a) = 30 (24 * 2a + a * 2a) = 30 (48a + 2a2) = 30 Solving 48a + 2a2 = 30 Solving for variable 'a'. Reorder the terms: -30 + 48a + 2a2 = 30 + -30 Combine like terms: 30 + -30 = 0 -30 + 48a + 2a2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-15 + 24a + a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-15 + 24a + a2)' equal to zero and attempt to solve: Simplifying -15 + 24a + a2 = 0 Solving -15 + 24a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '15' to each side of the equation. -15 + 24a + 15 + a2 = 0 + 15 Reorder the terms: -15 + 15 + 24a + a2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + 24a + a2 = 0 + 15 24a + a2 = 0 + 15 Combine like terms: 0 + 15 = 15 24a + a2 = 15 The a term is 24a. Take half its coefficient (12). Square it (144) and add it to both sides. Add '144' to each side of the equation. 24a + 144 + a2 = 15 + 144 Reorder the terms: 144 + 24a + a2 = 15 + 144 Combine like terms: 15 + 144 = 159 144 + 24a + a2 = 159 Factor a perfect square on the left side: (a + 12)(a + 12) = 159 Calculate the square root of the right side: 12.609520213 Break this problem into two subproblems by setting (a + 12) equal to 12.609520213 and -12.609520213.Subproblem 1
a + 12 = 12.609520213 Simplifying a + 12 = 12.609520213 Reorder the terms: 12 + a = 12.609520213 Solving 12 + a = 12.609520213 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + a = 12.609520213 + -12 Combine like terms: 12 + -12 = 0 0 + a = 12.609520213 + -12 a = 12.609520213 + -12 Combine like terms: 12.609520213 + -12 = 0.609520213 a = 0.609520213 Simplifying a = 0.609520213Subproblem 2
a + 12 = -12.609520213 Simplifying a + 12 = -12.609520213 Reorder the terms: 12 + a = -12.609520213 Solving 12 + a = -12.609520213 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + a = -12.609520213 + -12 Combine like terms: 12 + -12 = 0 0 + a = -12.609520213 + -12 a = -12.609520213 + -12 Combine like terms: -12.609520213 + -12 = -24.609520213 a = -24.609520213 Simplifying a = -24.609520213Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.609520213, -24.609520213}Solution
a = {0.609520213, -24.609520213}
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