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Simplifying (12a * a + 3a + -6) + (-4a * a + -5a + -8) = 0 Multiply a * a (12a2 + 3a + -6) + (-4a * a + -5a + -8) = 0 Reorder the terms: (-6 + 3a + 12a2) + (-4a * a + -5a + -8) = 0 Remove parenthesis around (-6 + 3a + 12a2) -6 + 3a + 12a2 + (-4a * a + -5a + -8) = 0 Multiply a * a -6 + 3a + 12a2 + (-4a2 + -5a + -8) = 0 Reorder the terms: -6 + 3a + 12a2 + (-8 + -5a + -4a2) = 0 Remove parenthesis around (-8 + -5a + -4a2) -6 + 3a + 12a2 + -8 + -5a + -4a2 = 0 Reorder the terms: -6 + -8 + 3a + -5a + 12a2 + -4a2 = 0 Combine like terms: -6 + -8 = -14 -14 + 3a + -5a + 12a2 + -4a2 = 0 Combine like terms: 3a + -5a = -2a -14 + -2a + 12a2 + -4a2 = 0 Combine like terms: 12a2 + -4a2 = 8a2 -14 + -2a + 8a2 = 0 Solving -14 + -2a + 8a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + -1a + 4a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + -1a + 4a2)' equal to zero and attempt to solve: Simplifying -7 + -1a + 4a2 = 0 Solving -7 + -1a + 4a2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.75 + -0.25a + a2 = 0 Move the constant term to the right: Add '1.75' to each side of the equation. -1.75 + -0.25a + 1.75 + a2 = 0 + 1.75 Reorder the terms: -1.75 + 1.75 + -0.25a + a2 = 0 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + -0.25a + a2 = 0 + 1.75 -0.25a + a2 = 0 + 1.75 Combine like terms: 0 + 1.75 = 1.75 -0.25a + a2 = 1.75 The a term is -0.25a. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25a + 0.015625 + a2 = 1.75 + 0.015625 Reorder the terms: 0.015625 + -0.25a + a2 = 1.75 + 0.015625 Combine like terms: 1.75 + 0.015625 = 1.765625 0.015625 + -0.25a + a2 = 1.765625 Factor a perfect square on the left side: (a + -0.125)(a + -0.125) = 1.765625 Calculate the square root of the right side: 1.328768227 Break this problem into two subproblems by setting (a + -0.125) equal to 1.328768227 and -1.328768227.Subproblem 1
a + -0.125 = 1.328768227 Simplifying a + -0.125 = 1.328768227 Reorder the terms: -0.125 + a = 1.328768227 Solving -0.125 + a = 1.328768227 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + a = 1.328768227 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + a = 1.328768227 + 0.125 a = 1.328768227 + 0.125 Combine like terms: 1.328768227 + 0.125 = 1.453768227 a = 1.453768227 Simplifying a = 1.453768227Subproblem 2
a + -0.125 = -1.328768227 Simplifying a + -0.125 = -1.328768227 Reorder the terms: -0.125 + a = -1.328768227 Solving -0.125 + a = -1.328768227 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + a = -1.328768227 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + a = -1.328768227 + 0.125 a = -1.328768227 + 0.125 Combine like terms: -1.328768227 + 0.125 = -1.203768227 a = -1.203768227 Simplifying a = -1.203768227Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.453768227, -1.203768227}Solution
a = {1.453768227, -1.203768227}
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