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Simplifying (9a2 + 7a + -9) + (2a2 + 8a + 12) = 0 Reorder the terms: (-9 + 7a + 9a2) + (2a2 + 8a + 12) = 0 Remove parenthesis around (-9 + 7a + 9a2) -9 + 7a + 9a2 + (2a2 + 8a + 12) = 0 Reorder the terms: -9 + 7a + 9a2 + (12 + 8a + 2a2) = 0 Remove parenthesis around (12 + 8a + 2a2) -9 + 7a + 9a2 + 12 + 8a + 2a2 = 0 Reorder the terms: -9 + 12 + 7a + 8a + 9a2 + 2a2 = 0 Combine like terms: -9 + 12 = 3 3 + 7a + 8a + 9a2 + 2a2 = 0 Combine like terms: 7a + 8a = 15a 3 + 15a + 9a2 + 2a2 = 0 Combine like terms: 9a2 + 2a2 = 11a2 3 + 15a + 11a2 = 0 Solving 3 + 15a + 11a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. 0.2727272727 + 1.363636364a + a2 = 0 Move the constant term to the right: Add '-0.2727272727' to each side of the equation. 0.2727272727 + 1.363636364a + -0.2727272727 + a2 = 0 + -0.2727272727 Reorder the terms: 0.2727272727 + -0.2727272727 + 1.363636364a + a2 = 0 + -0.2727272727 Combine like terms: 0.2727272727 + -0.2727272727 = 0.0000000000 0.0000000000 + 1.363636364a + a2 = 0 + -0.2727272727 1.363636364a + a2 = 0 + -0.2727272727 Combine like terms: 0 + -0.2727272727 = -0.2727272727 1.363636364a + a2 = -0.2727272727 The a term is 1.363636364a. Take half its coefficient (0.681818182). Square it (0.4648760333) and add it to both sides. Add '0.4648760333' to each side of the equation. 1.363636364a + 0.4648760333 + a2 = -0.2727272727 + 0.4648760333 Reorder the terms: 0.4648760333 + 1.363636364a + a2 = -0.2727272727 + 0.4648760333 Combine like terms: -0.2727272727 + 0.4648760333 = 0.1921487606 0.4648760333 + 1.363636364a + a2 = 0.1921487606 Factor a perfect square on the left side: (a + 0.681818182)(a + 0.681818182) = 0.1921487606 Calculate the square root of the right side: 0.438347762 Break this problem into two subproblems by setting (a + 0.681818182) equal to 0.438347762 and -0.438347762.Subproblem 1
a + 0.681818182 = 0.438347762 Simplifying a + 0.681818182 = 0.438347762 Reorder the terms: 0.681818182 + a = 0.438347762 Solving 0.681818182 + a = 0.438347762 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.681818182' to each side of the equation. 0.681818182 + -0.681818182 + a = 0.438347762 + -0.681818182 Combine like terms: 0.681818182 + -0.681818182 = 0.000000000 0.000000000 + a = 0.438347762 + -0.681818182 a = 0.438347762 + -0.681818182 Combine like terms: 0.438347762 + -0.681818182 = -0.24347042 a = -0.24347042 Simplifying a = -0.24347042Subproblem 2
a + 0.681818182 = -0.438347762 Simplifying a + 0.681818182 = -0.438347762 Reorder the terms: 0.681818182 + a = -0.438347762 Solving 0.681818182 + a = -0.438347762 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.681818182' to each side of the equation. 0.681818182 + -0.681818182 + a = -0.438347762 + -0.681818182 Combine like terms: 0.681818182 + -0.681818182 = 0.000000000 0.000000000 + a = -0.438347762 + -0.681818182 a = -0.438347762 + -0.681818182 Combine like terms: -0.438347762 + -0.681818182 = -1.120165944 a = -1.120165944 Simplifying a = -1.120165944Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.24347042, -1.120165944}
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