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Simplifying (2a + -3)(a + -1) + -2a + -4 = 0 Reorder the terms: (-3 + 2a)(a + -1) + -2a + -4 = 0 Reorder the terms: (-3 + 2a)(-1 + a) + -2a + -4 = 0 Multiply (-3 + 2a) * (-1 + a) (-3(-1 + a) + 2a * (-1 + a)) + -2a + -4 = 0 ((-1 * -3 + a * -3) + 2a * (-1 + a)) + -2a + -4 = 0 ((3 + -3a) + 2a * (-1 + a)) + -2a + -4 = 0 (3 + -3a + (-1 * 2a + a * 2a)) + -2a + -4 = 0 (3 + -3a + (-2a + 2a2)) + -2a + -4 = 0 Combine like terms: -3a + -2a = -5a (3 + -5a + 2a2) + -2a + -4 = 0 Reorder the terms: 3 + -4 + -5a + -2a + 2a2 = 0 Combine like terms: 3 + -4 = -1 -1 + -5a + -2a + 2a2 = 0 Combine like terms: -5a + -2a = -7a -1 + -7a + 2a2 = 0 Solving -1 + -7a + 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'. -0.5 + -3.5a + a2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -3.5a + 0.5 + a2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -3.5a + a2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -3.5a + a2 = 0 + 0.5 -3.5a + a2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -3.5a + a2 = 0.5 The a term is -3.5a. Take half its coefficient (-1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. -3.5a + 3.0625 + a2 = 0.5 + 3.0625 Reorder the terms: 3.0625 + -3.5a + a2 = 0.5 + 3.0625 Combine like terms: 0.5 + 3.0625 = 3.5625 3.0625 + -3.5a + a2 = 3.5625 Factor a perfect square on the left side: (a + -1.75)(a + -1.75) = 3.5625 Calculate the square root of the right side: 1.887458609 Break this problem into two subproblems by setting (a + -1.75) equal to 1.887458609 and -1.887458609.Subproblem 1
a + -1.75 = 1.887458609 Simplifying a + -1.75 = 1.887458609 Reorder the terms: -1.75 + a = 1.887458609 Solving -1.75 + a = 1.887458609 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + a = 1.887458609 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + a = 1.887458609 + 1.75 a = 1.887458609 + 1.75 Combine like terms: 1.887458609 + 1.75 = 3.637458609 a = 3.637458609 Simplifying a = 3.637458609Subproblem 2
a + -1.75 = -1.887458609 Simplifying a + -1.75 = -1.887458609 Reorder the terms: -1.75 + a = -1.887458609 Solving -1.75 + a = -1.887458609 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + a = -1.887458609 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + a = -1.887458609 + 1.75 a = -1.887458609 + 1.75 Combine like terms: -1.887458609 + 1.75 = -0.137458609 a = -0.137458609 Simplifying a = -0.137458609Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.637458609, -0.137458609}
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