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Simplifying (5a2 + 6a + 2) + -1(7a2 + -7a + 5) = 0 Reorder the terms: (2 + 6a + 5a2) + -1(7a2 + -7a + 5) = 0 Remove parenthesis around (2 + 6a + 5a2) 2 + 6a + 5a2 + -1(7a2 + -7a + 5) = 0 Reorder the terms: 2 + 6a + 5a2 + -1(5 + -7a + 7a2) = 0 2 + 6a + 5a2 + (5 * -1 + -7a * -1 + 7a2 * -1) = 0 2 + 6a + 5a2 + (-5 + 7a + -7a2) = 0 Reorder the terms: 2 + -5 + 6a + 7a + 5a2 + -7a2 = 0 Combine like terms: 2 + -5 = -3 -3 + 6a + 7a + 5a2 + -7a2 = 0 Combine like terms: 6a + 7a = 13a -3 + 13a + 5a2 + -7a2 = 0 Combine like terms: 5a2 + -7a2 = -2a2 -3 + 13a + -2a2 = 0 Solving -3 + 13a + -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'. 1.5 + -6.5a + a2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -6.5a + -1.5 + a2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -6.5a + a2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -6.5a + a2 = 0 + -1.5 -6.5a + a2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -6.5a + a2 = -1.5 The a term is -6.5a. Take half its coefficient (-3.25). Square it (10.5625) and add it to both sides. Add '10.5625' to each side of the equation. -6.5a + 10.5625 + a2 = -1.5 + 10.5625 Reorder the terms: 10.5625 + -6.5a + a2 = -1.5 + 10.5625 Combine like terms: -1.5 + 10.5625 = 9.0625 10.5625 + -6.5a + a2 = 9.0625 Factor a perfect square on the left side: (a + -3.25)(a + -3.25) = 9.0625 Calculate the square root of the right side: 3.010398645 Break this problem into two subproblems by setting (a + -3.25) equal to 3.010398645 and -3.010398645.Subproblem 1
a + -3.25 = 3.010398645 Simplifying a + -3.25 = 3.010398645 Reorder the terms: -3.25 + a = 3.010398645 Solving -3.25 + a = 3.010398645 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + a = 3.010398645 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + a = 3.010398645 + 3.25 a = 3.010398645 + 3.25 Combine like terms: 3.010398645 + 3.25 = 6.260398645 a = 6.260398645 Simplifying a = 6.260398645Subproblem 2
a + -3.25 = -3.010398645 Simplifying a + -3.25 = -3.010398645 Reorder the terms: -3.25 + a = -3.010398645 Solving -3.25 + a = -3.010398645 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + a = -3.010398645 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + a = -3.010398645 + 3.25 a = -3.010398645 + 3.25 Combine like terms: -3.010398645 + 3.25 = 0.239601355 a = 0.239601355 Simplifying a = 0.239601355Solution
The solution to the problem is based on the solutions from the subproblems. a = {6.260398645, 0.239601355}
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