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Simplifying 12a2 + -36a = 15 Reorder the terms: -36a + 12a2 = 15 Solving -36a + 12a2 = 15 Solving for variable 'a'. Reorder the terms: -15 + -36a + 12a2 = 15 + -15 Combine like terms: 15 + -15 = 0 -15 + -36a + 12a2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-5 + -12a + 4a2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-5 + -12a + 4a2)' equal to zero and attempt to solve: Simplifying -5 + -12a + 4a2 = 0 Solving -5 + -12a + 4a2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.25 + -3a + a2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + -3a + 1.25 + a2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + -3a + a2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + -3a + a2 = 0 + 1.25 -3a + a2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 -3a + a2 = 1.25 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 = 1.25 + 2.25 Reorder the terms: 2.25 + -3a + a2 = 1.25 + 2.25 Combine like terms: 1.25 + 2.25 = 3.5 2.25 + -3a + a2 = 3.5 Factor a perfect square on the left side: (a + -1.5)(a + -1.5) = 3.5 Calculate the square root of the right side: 1.870828693 Break this problem into two subproblems by setting (a + -1.5) equal to 1.870828693 and -1.870828693.Subproblem 1
a + -1.5 = 1.870828693 Simplifying a + -1.5 = 1.870828693 Reorder the terms: -1.5 + a = 1.870828693 Solving -1.5 + a = 1.870828693 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 = 1.870828693 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + a = 1.870828693 + 1.5 a = 1.870828693 + 1.5 Combine like terms: 1.870828693 + 1.5 = 3.370828693 a = 3.370828693 Simplifying a = 3.370828693Subproblem 2
a + -1.5 = -1.870828693 Simplifying a + -1.5 = -1.870828693 Reorder the terms: -1.5 + a = -1.870828693 Solving -1.5 + a = -1.870828693 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 = -1.870828693 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + a = -1.870828693 + 1.5 a = -1.870828693 + 1.5 Combine like terms: -1.870828693 + 1.5 = -0.370828693 a = -0.370828693 Simplifying a = -0.370828693Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.370828693, -0.370828693}Solution
a = {3.370828693, -0.370828693}
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