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Simplifying 18a2 + -108a + 9 = 0 Reorder the terms: 9 + -108a + 18a2 = 0 Solving 9 + -108a + 18a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '9'. 9(1 + -12a + 2a2) = 0 Ignore the factor 9.Subproblem 1
Set the factor '(1 + -12a + 2a2)' equal to zero and attempt to solve: Simplifying 1 + -12a + 2a2 = 0 Solving 1 + -12a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.5 + -6a + a2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -6a + -0.5 + a2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -6a + a2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -6a + a2 = 0 + -0.5 -6a + a2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -6a + a2 = -0.5 The a term is -6a. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6a + 9 + a2 = -0.5 + 9 Reorder the terms: 9 + -6a + a2 = -0.5 + 9 Combine like terms: -0.5 + 9 = 8.5 9 + -6a + a2 = 8.5 Factor a perfect square on the left side: (a + -3)(a + -3) = 8.5 Calculate the square root of the right side: 2.915475947 Break this problem into two subproblems by setting (a + -3) equal to 2.915475947 and -2.915475947.Subproblem 1
a + -3 = 2.915475947 Simplifying a + -3 = 2.915475947 Reorder the terms: -3 + a = 2.915475947 Solving -3 + a = 2.915475947 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + a = 2.915475947 + 3 Combine like terms: -3 + 3 = 0 0 + a = 2.915475947 + 3 a = 2.915475947 + 3 Combine like terms: 2.915475947 + 3 = 5.915475947 a = 5.915475947 Simplifying a = 5.915475947Subproblem 2
a + -3 = -2.915475947 Simplifying a + -3 = -2.915475947 Reorder the terms: -3 + a = -2.915475947 Solving -3 + a = -2.915475947 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + a = -2.915475947 + 3 Combine like terms: -3 + 3 = 0 0 + a = -2.915475947 + 3 a = -2.915475947 + 3 Combine like terms: -2.915475947 + 3 = 0.084524053 a = 0.084524053 Simplifying a = 0.084524053Solution
The solution to the problem is based on the solutions from the subproblems. a = {5.915475947, 0.084524053}Solution
a = {5.915475947, 0.084524053}
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