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Simplifying 12(a + -3) + -2a(a + -4) = 0 Reorder the terms: 12(-3 + a) + -2a(a + -4) = 0 (-3 * 12 + a * 12) + -2a(a + -4) = 0 (-36 + 12a) + -2a(a + -4) = 0 Reorder the terms: -36 + 12a + -2a(-4 + a) = 0 -36 + 12a + (-4 * -2a + a * -2a) = 0 -36 + 12a + (8a + -2a2) = 0 Combine like terms: 12a + 8a = 20a -36 + 20a + -2a2 = 0 Solving -36 + 20a + -2a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '2'. 2(-18 + 10a + -1a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-18 + 10a + -1a2)' equal to zero and attempt to solve: Simplifying -18 + 10a + -1a2 = 0 Solving -18 + 10a + -1a2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 18 + -10a + a2 = 0 Move the constant term to the right: Add '-18' to each side of the equation. 18 + -10a + -18 + a2 = 0 + -18 Reorder the terms: 18 + -18 + -10a + a2 = 0 + -18 Combine like terms: 18 + -18 = 0 0 + -10a + a2 = 0 + -18 -10a + a2 = 0 + -18 Combine like terms: 0 + -18 = -18 -10a + a2 = -18 The a term is -10a. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10a + 25 + a2 = -18 + 25 Reorder the terms: 25 + -10a + a2 = -18 + 25 Combine like terms: -18 + 25 = 7 25 + -10a + a2 = 7 Factor a perfect square on the left side: (a + -5)(a + -5) = 7 Calculate the square root of the right side: 2.645751311 Break this problem into two subproblems by setting (a + -5) equal to 2.645751311 and -2.645751311.Subproblem 1
a + -5 = 2.645751311 Simplifying a + -5 = 2.645751311 Reorder the terms: -5 + a = 2.645751311 Solving -5 + a = 2.645751311 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + a = 2.645751311 + 5 Combine like terms: -5 + 5 = 0 0 + a = 2.645751311 + 5 a = 2.645751311 + 5 Combine like terms: 2.645751311 + 5 = 7.645751311 a = 7.645751311 Simplifying a = 7.645751311Subproblem 2
a + -5 = -2.645751311 Simplifying a + -5 = -2.645751311 Reorder the terms: -5 + a = -2.645751311 Solving -5 + a = -2.645751311 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + a = -2.645751311 + 5 Combine like terms: -5 + 5 = 0 0 + a = -2.645751311 + 5 a = -2.645751311 + 5 Combine like terms: -2.645751311 + 5 = 2.354248689 a = 2.354248689 Simplifying a = 2.354248689Solution
The solution to the problem is based on the solutions from the subproblems. a = {7.645751311, 2.354248689}Solution
a = {7.645751311, 2.354248689}
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