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Simplifying (a2 + -4a + 6) + (-3a2 + 13a + 1) = 0 Reorder the terms: (6 + -4a + a2) + (-3a2 + 13a + 1) = 0 Remove parenthesis around (6 + -4a + a2) 6 + -4a + a2 + (-3a2 + 13a + 1) = 0 Reorder the terms: 6 + -4a + a2 + (1 + 13a + -3a2) = 0 Remove parenthesis around (1 + 13a + -3a2) 6 + -4a + a2 + 1 + 13a + -3a2 = 0 Reorder the terms: 6 + 1 + -4a + 13a + a2 + -3a2 = 0 Combine like terms: 6 + 1 = 7 7 + -4a + 13a + a2 + -3a2 = 0 Combine like terms: -4a + 13a = 9a 7 + 9a + a2 + -3a2 = 0 Combine like terms: a2 + -3a2 = -2a2 7 + 9a + -2a2 = 0 Solving 7 + 9a + -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'. -3.5 + -4.5a + a2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + -4.5a + 3.5 + a2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + -4.5a + a2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + -4.5a + a2 = 0 + 3.5 -4.5a + a2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 -4.5a + a2 = 3.5 The a term is -4.5a. Take half its coefficient (-2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. -4.5a + 5.0625 + a2 = 3.5 + 5.0625 Reorder the terms: 5.0625 + -4.5a + a2 = 3.5 + 5.0625 Combine like terms: 3.5 + 5.0625 = 8.5625 5.0625 + -4.5a + a2 = 8.5625 Factor a perfect square on the left side: (a + -2.25)(a + -2.25) = 8.5625 Calculate the square root of the right side: 2.926174978 Break this problem into two subproblems by setting (a + -2.25) equal to 2.926174978 and -2.926174978.Subproblem 1
a + -2.25 = 2.926174978 Simplifying a + -2.25 = 2.926174978 Reorder the terms: -2.25 + a = 2.926174978 Solving -2.25 + a = 2.926174978 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + a = 2.926174978 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + a = 2.926174978 + 2.25 a = 2.926174978 + 2.25 Combine like terms: 2.926174978 + 2.25 = 5.176174978 a = 5.176174978 Simplifying a = 5.176174978Subproblem 2
a + -2.25 = -2.926174978 Simplifying a + -2.25 = -2.926174978 Reorder the terms: -2.25 + a = -2.926174978 Solving -2.25 + a = -2.926174978 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + a = -2.926174978 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + a = -2.926174978 + 2.25 a = -2.926174978 + 2.25 Combine like terms: -2.926174978 + 2.25 = -0.676174978 a = -0.676174978 Simplifying a = -0.676174978Solution
The solution to the problem is based on the solutions from the subproblems. a = {5.176174978, -0.676174978}
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