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Simplifying -4a2 + 18a + -15 = -7a2 + 9a Reorder the terms: -15 + 18a + -4a2 = -7a2 + 9a Reorder the terms: -15 + 18a + -4a2 = 9a + -7a2 Solving -15 + 18a + -4a2 = 9a + -7a2 Solving for variable 'a'. Reorder the terms: -15 + 18a + -9a + -4a2 + 7a2 = 9a + -7a2 + -9a + 7a2 Combine like terms: 18a + -9a = 9a -15 + 9a + -4a2 + 7a2 = 9a + -7a2 + -9a + 7a2 Combine like terms: -4a2 + 7a2 = 3a2 -15 + 9a + 3a2 = 9a + -7a2 + -9a + 7a2 Reorder the terms: -15 + 9a + 3a2 = 9a + -9a + -7a2 + 7a2 Combine like terms: 9a + -9a = 0 -15 + 9a + 3a2 = 0 + -7a2 + 7a2 -15 + 9a + 3a2 = -7a2 + 7a2 Combine like terms: -7a2 + 7a2 = 0 -15 + 9a + 3a2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-5 + 3a + a2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-5 + 3a + a2)' equal to zero and attempt to solve: Simplifying -5 + 3a + a2 = 0 Solving -5 + 3a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '5' to each side of the equation. -5 + 3a + 5 + a2 = 0 + 5 Reorder the terms: -5 + 5 + 3a + a2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 3a + a2 = 0 + 5 3a + a2 = 0 + 5 Combine like terms: 0 + 5 = 5 3a + a2 = 5 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 = 5 + 2.25 Reorder the terms: 2.25 + 3a + a2 = 5 + 2.25 Combine like terms: 5 + 2.25 = 7.25 2.25 + 3a + a2 = 7.25 Factor a perfect square on the left side: (a + 1.5)(a + 1.5) = 7.25 Calculate the square root of the right side: 2.692582404 Break this problem into two subproblems by setting (a + 1.5) equal to 2.692582404 and -2.692582404.Subproblem 1
a + 1.5 = 2.692582404 Simplifying a + 1.5 = 2.692582404 Reorder the terms: 1.5 + a = 2.692582404 Solving 1.5 + a = 2.692582404 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 = 2.692582404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = 2.692582404 + -1.5 a = 2.692582404 + -1.5 Combine like terms: 2.692582404 + -1.5 = 1.192582404 a = 1.192582404 Simplifying a = 1.192582404Subproblem 2
a + 1.5 = -2.692582404 Simplifying a + 1.5 = -2.692582404 Reorder the terms: 1.5 + a = -2.692582404 Solving 1.5 + a = -2.692582404 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 = -2.692582404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = -2.692582404 + -1.5 a = -2.692582404 + -1.5 Combine like terms: -2.692582404 + -1.5 = -4.192582404 a = -4.192582404 Simplifying a = -4.192582404Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.192582404, -4.192582404}Solution
a = {1.192582404, -4.192582404}
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