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Simplifying 6a2 + 5a + -8a2 + 5 = 0 Reorder the terms: 5 + 5a + 6a2 + -8a2 = 0 Combine like terms: 6a2 + -8a2 = -2a2 5 + 5a + -2a2 = 0 Solving 5 + 5a + -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 + -2.5a + a2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + -2.5a + 2.5 + a2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + -2.5a + a2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + -2.5a + a2 = 0 + 2.5 -2.5a + a2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 -2.5a + a2 = 2.5 The a term is -2.5a. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5a + 1.5625 + a2 = 2.5 + 1.5625 Reorder the terms: 1.5625 + -2.5a + a2 = 2.5 + 1.5625 Combine like terms: 2.5 + 1.5625 = 4.0625 1.5625 + -2.5a + a2 = 4.0625 Factor a perfect square on the left side: (a + -1.25)(a + -1.25) = 4.0625 Calculate the square root of the right side: 2.015564437 Break this problem into two subproblems by setting (a + -1.25) equal to 2.015564437 and -2.015564437.Subproblem 1
a + -1.25 = 2.015564437 Simplifying a + -1.25 = 2.015564437 Reorder the terms: -1.25 + a = 2.015564437 Solving -1.25 + a = 2.015564437 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + a = 2.015564437 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + a = 2.015564437 + 1.25 a = 2.015564437 + 1.25 Combine like terms: 2.015564437 + 1.25 = 3.265564437 a = 3.265564437 Simplifying a = 3.265564437Subproblem 2
a + -1.25 = -2.015564437 Simplifying a + -1.25 = -2.015564437 Reorder the terms: -1.25 + a = -2.015564437 Solving -1.25 + a = -2.015564437 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + a = -2.015564437 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + a = -2.015564437 + 1.25 a = -2.015564437 + 1.25 Combine like terms: -2.015564437 + 1.25 = -0.765564437 a = -0.765564437 Simplifying a = -0.765564437Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.265564437, -0.765564437}
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