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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 = 7 7 + -6a + 7a + 5a2 + -7a2 = 0 Combine like terms: -6a + 7a = 1a 7 + 1a + 5a2 + -7a2 = 0 Combine like terms: 5a2 + -7a2 = -2a2 7 + 1a + -2a2 = 0 Solving 7 + 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'. -3.5 + -0.5a + a2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + -0.5a + 3.5 + a2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + -0.5a + a2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + -0.5a + a2 = 0 + 3.5 -0.5a + a2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 -0.5a + a2 = 3.5 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 = 3.5 + 0.0625 Reorder the terms: 0.0625 + -0.5a + a2 = 3.5 + 0.0625 Combine like terms: 3.5 + 0.0625 = 3.5625 0.0625 + -0.5a + a2 = 3.5625 Factor a perfect square on the left side: (a + -0.25)(a + -0.25) = 3.5625 Calculate the square root of the right side: 1.887458609 Break this problem into two subproblems by setting (a + -0.25) equal to 1.887458609 and -1.887458609.Subproblem 1
a + -0.25 = 1.887458609 Simplifying a + -0.25 = 1.887458609 Reorder the terms: -0.25 + a = 1.887458609 Solving -0.25 + a = 1.887458609 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 = 1.887458609 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = 1.887458609 + 0.25 a = 1.887458609 + 0.25 Combine like terms: 1.887458609 + 0.25 = 2.137458609 a = 2.137458609 Simplifying a = 2.137458609Subproblem 2
a + -0.25 = -1.887458609 Simplifying a + -0.25 = -1.887458609 Reorder the terms: -0.25 + a = -1.887458609 Solving -0.25 + a = -1.887458609 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 = -1.887458609 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = -1.887458609 + 0.25 a = -1.887458609 + 0.25 Combine like terms: -1.887458609 + 0.25 = -1.637458609 a = -1.637458609 Simplifying a = -1.637458609Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.137458609, -1.637458609}
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