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Simplifying 9a2 + 18a + -12 = 0 Reorder the terms: -12 + 18a + 9a2 = 0 Solving -12 + 18a + 9a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '3'. 3(-4 + 6a + 3a2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-4 + 6a + 3a2)' equal to zero and attempt to solve: Simplifying -4 + 6a + 3a2 = 0 Solving -4 + 6a + 3a2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1.333333333 + 2a + a2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + 2a + 1.333333333 + a2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + 2a + a2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + 2a + a2 = 0 + 1.333333333 2a + a2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 2a + a2 = 1.333333333 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 = 1.333333333 + 1 Reorder the terms: 1 + 2a + a2 = 1.333333333 + 1 Combine like terms: 1.333333333 + 1 = 2.333333333 1 + 2a + a2 = 2.333333333 Factor a perfect square on the left side: (a + 1)(a + 1) = 2.333333333 Calculate the square root of the right side: 1.527525232 Break this problem into two subproblems by setting (a + 1) equal to 1.527525232 and -1.527525232.Subproblem 1
a + 1 = 1.527525232 Simplifying a + 1 = 1.527525232 Reorder the terms: 1 + a = 1.527525232 Solving 1 + a = 1.527525232 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 = 1.527525232 + -1 Combine like terms: 1 + -1 = 0 0 + a = 1.527525232 + -1 a = 1.527525232 + -1 Combine like terms: 1.527525232 + -1 = 0.527525232 a = 0.527525232 Simplifying a = 0.527525232Subproblem 2
a + 1 = -1.527525232 Simplifying a + 1 = -1.527525232 Reorder the terms: 1 + a = -1.527525232 Solving 1 + a = -1.527525232 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 = -1.527525232 + -1 Combine like terms: 1 + -1 = 0 0 + a = -1.527525232 + -1 a = -1.527525232 + -1 Combine like terms: -1.527525232 + -1 = -2.527525232 a = -2.527525232 Simplifying a = -2.527525232Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.527525232, -2.527525232}Solution
a = {0.527525232, -2.527525232}
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