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Simplifying (2a + -3)(a + -1) + -4a + -4 = 0 Reorder the terms: (-3 + 2a)(a + -1) + -4a + -4 = 0 Reorder the terms: (-3 + 2a)(-1 + a) + -4a + -4 = 0 Multiply (-3 + 2a) * (-1 + a) (-3(-1 + a) + 2a * (-1 + a)) + -4a + -4 = 0 ((-1 * -3 + a * -3) + 2a * (-1 + a)) + -4a + -4 = 0 ((3 + -3a) + 2a * (-1 + a)) + -4a + -4 = 0 (3 + -3a + (-1 * 2a + a * 2a)) + -4a + -4 = 0 (3 + -3a + (-2a + 2a2)) + -4a + -4 = 0 Combine like terms: -3a + -2a = -5a (3 + -5a + 2a2) + -4a + -4 = 0 Reorder the terms: 3 + -4 + -5a + -4a + 2a2 = 0 Combine like terms: 3 + -4 = -1 -1 + -5a + -4a + 2a2 = 0 Combine like terms: -5a + -4a = -9a -1 + -9a + 2a2 = 0 Solving -1 + -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'. -0.5 + -4.5a + a2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -4.5a + 0.5 + a2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -4.5a + a2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -4.5a + a2 = 0 + 0.5 -4.5a + a2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -4.5a + a2 = 0.5 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 = 0.5 + 5.0625 Reorder the terms: 5.0625 + -4.5a + a2 = 0.5 + 5.0625 Combine like terms: 0.5 + 5.0625 = 5.5625 5.0625 + -4.5a + a2 = 5.5625 Factor a perfect square on the left side: (a + -2.25)(a + -2.25) = 5.5625 Calculate the square root of the right side: 2.358495283 Break this problem into two subproblems by setting (a + -2.25) equal to 2.358495283 and -2.358495283.Subproblem 1
a + -2.25 = 2.358495283 Simplifying a + -2.25 = 2.358495283 Reorder the terms: -2.25 + a = 2.358495283 Solving -2.25 + a = 2.358495283 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 = 2.358495283 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + a = 2.358495283 + 2.25 a = 2.358495283 + 2.25 Combine like terms: 2.358495283 + 2.25 = 4.608495283 a = 4.608495283 Simplifying a = 4.608495283Subproblem 2
a + -2.25 = -2.358495283 Simplifying a + -2.25 = -2.358495283 Reorder the terms: -2.25 + a = -2.358495283 Solving -2.25 + a = -2.358495283 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 = -2.358495283 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + a = -2.358495283 + 2.25 a = -2.358495283 + 2.25 Combine like terms: -2.358495283 + 2.25 = -0.108495283 a = -0.108495283 Simplifying a = -0.108495283Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.608495283, -0.108495283}
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