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Simplifying 12a2 + 45a + 20 = 2 Reorder the terms: 20 + 45a + 12a2 = 2 Solving 20 + 45a + 12a2 = 2 Solving for variable 'a'. Reorder the terms: 20 + -2 + 45a + 12a2 = 2 + -2 Combine like terms: 20 + -2 = 18 18 + 45a + 12a2 = 2 + -2 Combine like terms: 2 + -2 = 0 18 + 45a + 12a2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(6 + 15a + 4a2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(6 + 15a + 4a2)' equal to zero and attempt to solve: Simplifying 6 + 15a + 4a2 = 0 Solving 6 + 15a + 4a2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 1.5 + 3.75a + a2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + 3.75a + -1.5 + a2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + 3.75a + a2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + 3.75a + a2 = 0 + -1.5 3.75a + a2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 3.75a + a2 = -1.5 The a term is 3.75a. Take half its coefficient (1.875). Square it (3.515625) and add it to both sides. Add '3.515625' to each side of the equation. 3.75a + 3.515625 + a2 = -1.5 + 3.515625 Reorder the terms: 3.515625 + 3.75a + a2 = -1.5 + 3.515625 Combine like terms: -1.5 + 3.515625 = 2.015625 3.515625 + 3.75a + a2 = 2.015625 Factor a perfect square on the left side: (a + 1.875)(a + 1.875) = 2.015625 Calculate the square root of the right side: 1.419727086 Break this problem into two subproblems by setting (a + 1.875) equal to 1.419727086 and -1.419727086.Subproblem 1
a + 1.875 = 1.419727086 Simplifying a + 1.875 = 1.419727086 Reorder the terms: 1.875 + a = 1.419727086 Solving 1.875 + a = 1.419727086 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.875' to each side of the equation. 1.875 + -1.875 + a = 1.419727086 + -1.875 Combine like terms: 1.875 + -1.875 = 0.000 0.000 + a = 1.419727086 + -1.875 a = 1.419727086 + -1.875 Combine like terms: 1.419727086 + -1.875 = -0.455272914 a = -0.455272914 Simplifying a = -0.455272914Subproblem 2
a + 1.875 = -1.419727086 Simplifying a + 1.875 = -1.419727086 Reorder the terms: 1.875 + a = -1.419727086 Solving 1.875 + a = -1.419727086 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.875' to each side of the equation. 1.875 + -1.875 + a = -1.419727086 + -1.875 Combine like terms: 1.875 + -1.875 = 0.000 0.000 + a = -1.419727086 + -1.875 a = -1.419727086 + -1.875 Combine like terms: -1.419727086 + -1.875 = -3.294727086 a = -3.294727086 Simplifying a = -3.294727086Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.455272914, -3.294727086}Solution
a = {-0.455272914, -3.294727086}
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