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Simplifying -1a2 + 7a + 1 = -5a2 + -12a + -9 Reorder the terms: 1 + 7a + -1a2 = -5a2 + -12a + -9 Reorder the terms: 1 + 7a + -1a2 = -9 + -12a + -5a2 Solving 1 + 7a + -1a2 = -9 + -12a + -5a2 Solving for variable 'a'. Reorder the terms: 1 + 9 + 7a + 12a + -1a2 + 5a2 = -9 + -12a + -5a2 + 9 + 12a + 5a2 Combine like terms: 1 + 9 = 10 10 + 7a + 12a + -1a2 + 5a2 = -9 + -12a + -5a2 + 9 + 12a + 5a2 Combine like terms: 7a + 12a = 19a 10 + 19a + -1a2 + 5a2 = -9 + -12a + -5a2 + 9 + 12a + 5a2 Combine like terms: -1a2 + 5a2 = 4a2 10 + 19a + 4a2 = -9 + -12a + -5a2 + 9 + 12a + 5a2 Reorder the terms: 10 + 19a + 4a2 = -9 + 9 + -12a + 12a + -5a2 + 5a2 Combine like terms: -9 + 9 = 0 10 + 19a + 4a2 = 0 + -12a + 12a + -5a2 + 5a2 10 + 19a + 4a2 = -12a + 12a + -5a2 + 5a2 Combine like terms: -12a + 12a = 0 10 + 19a + 4a2 = 0 + -5a2 + 5a2 10 + 19a + 4a2 = -5a2 + 5a2 Combine like terms: -5a2 + 5a2 = 0 10 + 19a + 4a2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 2.5 + 4.75a + a2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + 4.75a + -2.5 + a2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + 4.75a + a2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + 4.75a + a2 = 0 + -2.5 4.75a + a2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 4.75a + a2 = -2.5 The a term is 4.75a. Take half its coefficient (2.375). Square it (5.640625) and add it to both sides. Add '5.640625' to each side of the equation. 4.75a + 5.640625 + a2 = -2.5 + 5.640625 Reorder the terms: 5.640625 + 4.75a + a2 = -2.5 + 5.640625 Combine like terms: -2.5 + 5.640625 = 3.140625 5.640625 + 4.75a + a2 = 3.140625 Factor a perfect square on the left side: (a + 2.375)(a + 2.375) = 3.140625 Calculate the square root of the right side: 1.77218086 Break this problem into two subproblems by setting (a + 2.375) equal to 1.77218086 and -1.77218086.Subproblem 1
a + 2.375 = 1.77218086 Simplifying a + 2.375 = 1.77218086 Reorder the terms: 2.375 + a = 1.77218086 Solving 2.375 + a = 1.77218086 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.375' to each side of the equation. 2.375 + -2.375 + a = 1.77218086 + -2.375 Combine like terms: 2.375 + -2.375 = 0.000 0.000 + a = 1.77218086 + -2.375 a = 1.77218086 + -2.375 Combine like terms: 1.77218086 + -2.375 = -0.60281914 a = -0.60281914 Simplifying a = -0.60281914Subproblem 2
a + 2.375 = -1.77218086 Simplifying a + 2.375 = -1.77218086 Reorder the terms: 2.375 + a = -1.77218086 Solving 2.375 + a = -1.77218086 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.375' to each side of the equation. 2.375 + -2.375 + a = -1.77218086 + -2.375 Combine like terms: 2.375 + -2.375 = 0.000 0.000 + a = -1.77218086 + -2.375 a = -1.77218086 + -2.375 Combine like terms: -1.77218086 + -2.375 = -4.14718086 a = -4.14718086 Simplifying a = -4.14718086Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.60281914, -4.14718086}
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