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Simplifying (-1a2 + 2) + -1(a2 + -4) + (6a2 + -5a * 3) = 0 Reorder the terms: (2 + -1a2) + -1(a2 + -4) + (6a2 + -5a * 3) = 0 Remove parenthesis around (2 + -1a2) 2 + -1a2 + -1(a2 + -4) + (6a2 + -5a * 3) = 0 Reorder the terms: 2 + -1a2 + -1(-4 + a2) + (6a2 + -5a * 3) = 0 2 + -1a2 + (-4 * -1 + a2 * -1) + (6a2 + -5a * 3) = 0 2 + -1a2 + (4 + -1a2) + (6a2 + -5a * 3) = 0 Reorder the terms for easier multiplication: 2 + -1a2 + 4 + -1a2 + (6a2 + -5 * 3a) = 0 Multiply -5 * 3 2 + -1a2 + 4 + -1a2 + (6a2 + -15a) = 0 Reorder the terms: 2 + -1a2 + 4 + -1a2 + (-15a + 6a2) = 0 Remove parenthesis around (-15a + 6a2) 2 + -1a2 + 4 + -1a2 + -15a + 6a2 = 0 Reorder the terms: 2 + 4 + -15a + -1a2 + -1a2 + 6a2 = 0 Combine like terms: 2 + 4 = 6 6 + -15a + -1a2 + -1a2 + 6a2 = 0 Combine like terms: -1a2 + -1a2 = -2a2 6 + -15a + -2a2 + 6a2 = 0 Combine like terms: -2a2 + 6a2 = 4a2 6 + -15a + 4a2 = 0 Solving 6 + -15a + 4a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 1.5 + -3.75a + a2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -3.75a + -1.5 + a2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -3.75a + a2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -3.75a + a2 = 0 + -1.5 -3.75a + a2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -3.75a + a2 = -1.5 The a term is -3.75a. Take half its coefficient (-1.875). Square it (3.515625) and add it to both sides. Add '3.515625' to each side of the equation. -3.75a + 3.515625 + a2 = -1.5 + 3.515625 Reorder the terms: 3.515625 + -3.75a + a2 = -1.5 + 3.515625 Combine like terms: -1.5 + 3.515625 = 2.015625 3.515625 + -3.75a + a2 = 2.015625 Factor a perfect square on the left side: (a + -1.875)(a + -1.875) = 2.015625 Calculate the square root of the right side: 1.419727086 Break this problem into two subproblems by setting (a + -1.875) equal to 1.419727086 and -1.419727086.Subproblem 1
a + -1.875 = 1.419727086 Simplifying a + -1.875 = 1.419727086 Reorder the terms: -1.875 + a = 1.419727086 Solving -1.875 + a = 1.419727086 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + a = 1.419727086 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + a = 1.419727086 + 1.875 a = 1.419727086 + 1.875 Combine like terms: 1.419727086 + 1.875 = 3.294727086 a = 3.294727086 Simplifying a = 3.294727086Subproblem 2
a + -1.875 = -1.419727086 Simplifying a + -1.875 = -1.419727086 Reorder the terms: -1.875 + a = -1.419727086 Solving -1.875 + a = -1.419727086 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + a = -1.419727086 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + a = -1.419727086 + 1.875 a = -1.419727086 + 1.875 Combine like terms: -1.419727086 + 1.875 = 0.455272914 a = 0.455272914 Simplifying a = 0.455272914Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.294727086, 0.455272914}
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