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Simplifying (6a2 + 5a + 10) + -1(4a2 + 6a + 12) = 0 Reorder the terms: (10 + 5a + 6a2) + -1(4a2 + 6a + 12) = 0 Remove parenthesis around (10 + 5a + 6a2) 10 + 5a + 6a2 + -1(4a2 + 6a + 12) = 0 Reorder the terms: 10 + 5a + 6a2 + -1(12 + 6a + 4a2) = 0 10 + 5a + 6a2 + (12 * -1 + 6a * -1 + 4a2 * -1) = 0 10 + 5a + 6a2 + (-12 + -6a + -4a2) = 0 Reorder the terms: 10 + -12 + 5a + -6a + 6a2 + -4a2 = 0 Combine like terms: 10 + -12 = -2 -2 + 5a + -6a + 6a2 + -4a2 = 0 Combine like terms: 5a + -6a = -1a -2 + -1a + 6a2 + -4a2 = 0 Combine like terms: 6a2 + -4a2 = 2a2 -2 + -1a + 2a2 = 0 Solving -2 + -1a + 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'. -1 + -0.5a + a2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.5a + 1 + a2 = 0 + 1 Reorder the terms: -1 + 1 + -0.5a + a2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.5a + a2 = 0 + 1 -0.5a + a2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.5a + a2 = 1 The a term is -0.5a. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5a + 0.0625 + a2 = 1 + 0.0625 Reorder the terms: 0.0625 + -0.5a + a2 = 1 + 0.0625 Combine like terms: 1 + 0.0625 = 1.0625 0.0625 + -0.5a + a2 = 1.0625 Factor a perfect square on the left side: (a + -0.25)(a + -0.25) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (a + -0.25) equal to 1.030776406 and -1.030776406.Subproblem 1
a + -0.25 = 1.030776406 Simplifying a + -0.25 = 1.030776406 Reorder the terms: -0.25 + a = 1.030776406 Solving -0.25 + a = 1.030776406 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + a = 1.030776406 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = 1.030776406 + 0.25 a = 1.030776406 + 0.25 Combine like terms: 1.030776406 + 0.25 = 1.280776406 a = 1.280776406 Simplifying a = 1.280776406Subproblem 2
a + -0.25 = -1.030776406 Simplifying a + -0.25 = -1.030776406 Reorder the terms: -0.25 + a = -1.030776406 Solving -0.25 + a = -1.030776406 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + a = -1.030776406 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = -1.030776406 + 0.25 a = -1.030776406 + 0.25 Combine like terms: -1.030776406 + 0.25 = -0.780776406 a = -0.780776406 Simplifying a = -0.780776406Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.280776406, -0.780776406}
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