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Simplifying 6a2 + 12a + -15 = 0 Reorder the terms: -15 + 12a + 6a2 = 0 Solving -15 + 12a + 6a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '3'. 3(-5 + 4a + 2a2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-5 + 4a + 2a2)' equal to zero and attempt to solve: Simplifying -5 + 4a + 2a2 = 0 Solving -5 + 4a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2.5 + 2a + a2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 2a + 2.5 + a2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 2a + a2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 2a + a2 = 0 + 2.5 2a + a2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 2a + a2 = 2.5 The a term is 2a. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2a + 1 + a2 = 2.5 + 1 Reorder the terms: 1 + 2a + a2 = 2.5 + 1 Combine like terms: 2.5 + 1 = 3.5 1 + 2a + a2 = 3.5 Factor a perfect square on the left side: (a + 1)(a + 1) = 3.5 Calculate the square root of the right side: 1.870828693 Break this problem into two subproblems by setting (a + 1) equal to 1.870828693 and -1.870828693.Subproblem 1
a + 1 = 1.870828693 Simplifying a + 1 = 1.870828693 Reorder the terms: 1 + a = 1.870828693 Solving 1 + a = 1.870828693 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + a = 1.870828693 + -1 Combine like terms: 1 + -1 = 0 0 + a = 1.870828693 + -1 a = 1.870828693 + -1 Combine like terms: 1.870828693 + -1 = 0.870828693 a = 0.870828693 Simplifying a = 0.870828693Subproblem 2
a + 1 = -1.870828693 Simplifying a + 1 = -1.870828693 Reorder the terms: 1 + a = -1.870828693 Solving 1 + a = -1.870828693 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + a = -1.870828693 + -1 Combine like terms: 1 + -1 = 0 0 + a = -1.870828693 + -1 a = -1.870828693 + -1 Combine like terms: -1.870828693 + -1 = -2.870828693 a = -2.870828693 Simplifying a = -2.870828693Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.870828693, -2.870828693}Solution
a = {0.870828693, -2.870828693}
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