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Simplifying (-1a2 + -5a + 3) + -1(a2 + 3a + -2) = 0 Reorder the terms: (3 + -5a + -1a2) + -1(a2 + 3a + -2) = 0 Remove parenthesis around (3 + -5a + -1a2) 3 + -5a + -1a2 + -1(a2 + 3a + -2) = 0 Reorder the terms: 3 + -5a + -1a2 + -1(-2 + 3a + a2) = 0 3 + -5a + -1a2 + (-2 * -1 + 3a * -1 + a2 * -1) = 0 3 + -5a + -1a2 + (2 + -3a + -1a2) = 0 Reorder the terms: 3 + 2 + -5a + -3a + -1a2 + -1a2 = 0 Combine like terms: 3 + 2 = 5 5 + -5a + -3a + -1a2 + -1a2 = 0 Combine like terms: -5a + -3a = -8a 5 + -8a + -1a2 + -1a2 = 0 Combine like terms: -1a2 + -1a2 = -2a2 5 + -8a + -2a2 = 0 Solving 5 + -8a + -2a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2.5 + 4a + a2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 4a + 2.5 + a2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 4a + a2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 4a + a2 = 0 + 2.5 4a + a2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 4a + a2 = 2.5 The a term is 4a. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4a + 4 + a2 = 2.5 + 4 Reorder the terms: 4 + 4a + a2 = 2.5 + 4 Combine like terms: 2.5 + 4 = 6.5 4 + 4a + a2 = 6.5 Factor a perfect square on the left side: (a + 2)(a + 2) = 6.5 Calculate the square root of the right side: 2.549509757 Break this problem into two subproblems by setting (a + 2) equal to 2.549509757 and -2.549509757.Subproblem 1
a + 2 = 2.549509757 Simplifying a + 2 = 2.549509757 Reorder the terms: 2 + a = 2.549509757 Solving 2 + a = 2.549509757 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + a = 2.549509757 + -2 Combine like terms: 2 + -2 = 0 0 + a = 2.549509757 + -2 a = 2.549509757 + -2 Combine like terms: 2.549509757 + -2 = 0.549509757 a = 0.549509757 Simplifying a = 0.549509757Subproblem 2
a + 2 = -2.549509757 Simplifying a + 2 = -2.549509757 Reorder the terms: 2 + a = -2.549509757 Solving 2 + a = -2.549509757 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + a = -2.549509757 + -2 Combine like terms: 2 + -2 = 0 0 + a = -2.549509757 + -2 a = -2.549509757 + -2 Combine like terms: -2.549509757 + -2 = -4.549509757 a = -4.549509757 Simplifying a = -4.549509757Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.549509757, -4.549509757}
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