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Simplifying (2a2 + 5a + 7) + -1(a2 + -3a + -1) = 0 Reorder the terms: (7 + 5a + 2a2) + -1(a2 + -3a + -1) = 0 Remove parenthesis around (7 + 5a + 2a2) 7 + 5a + 2a2 + -1(a2 + -3a + -1) = 0 Reorder the terms: 7 + 5a + 2a2 + -1(-1 + -3a + a2) = 0 7 + 5a + 2a2 + (-1 * -1 + -3a * -1 + a2 * -1) = 0 7 + 5a + 2a2 + (1 + 3a + -1a2) = 0 Reorder the terms: 7 + 1 + 5a + 3a + 2a2 + -1a2 = 0 Combine like terms: 7 + 1 = 8 8 + 5a + 3a + 2a2 + -1a2 = 0 Combine like terms: 5a + 3a = 8a 8 + 8a + 2a2 + -1a2 = 0 Combine like terms: 2a2 + -1a2 = 1a2 8 + 8a + 1a2 = 0 Solving 8 + 8a + 1a2 = 0 Solving for variable 'a'. Begin completing the square. Move the constant term to the right: Add '-8' to each side of the equation. 8 + 8a + -8 + a2 = 0 + -8 Reorder the terms: 8 + -8 + 8a + a2 = 0 + -8 Combine like terms: 8 + -8 = 0 0 + 8a + a2 = 0 + -8 8a + a2 = 0 + -8 Combine like terms: 0 + -8 = -8 8a + a2 = -8 The a term is 8a. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8a + 16 + a2 = -8 + 16 Reorder the terms: 16 + 8a + a2 = -8 + 16 Combine like terms: -8 + 16 = 8 16 + 8a + a2 = 8 Factor a perfect square on the left side: (a + 4)(a + 4) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (a + 4) equal to 2.828427125 and -2.828427125.Subproblem 1
a + 4 = 2.828427125 Simplifying a + 4 = 2.828427125 Reorder the terms: 4 + a = 2.828427125 Solving 4 + a = 2.828427125 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + a = 2.828427125 + -4 Combine like terms: 4 + -4 = 0 0 + a = 2.828427125 + -4 a = 2.828427125 + -4 Combine like terms: 2.828427125 + -4 = -1.171572875 a = -1.171572875 Simplifying a = -1.171572875Subproblem 2
a + 4 = -2.828427125 Simplifying a + 4 = -2.828427125 Reorder the terms: 4 + a = -2.828427125 Solving 4 + a = -2.828427125 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + a = -2.828427125 + -4 Combine like terms: 4 + -4 = 0 0 + a = -2.828427125 + -4 a = -2.828427125 + -4 Combine like terms: -2.828427125 + -4 = -6.828427125 a = -6.828427125 Simplifying a = -6.828427125Solution
The solution to the problem is based on the solutions from the subproblems. a = {-1.171572875, -6.828427125}
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