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Simplifying 7a + -4 + -2(a + 5) + 256(a + -3) + 2a = 4(a + -1) * 2a + -5 Reorder the terms: 7a + -4 + -2(5 + a) + 256(a + -3) + 2a = 4(a + -1) * 2a + -5 7a + -4 + (5 * -2 + a * -2) + 256(a + -3) + 2a = 4(a + -1) * 2a + -5 7a + -4 + (-10 + -2a) + 256(a + -3) + 2a = 4(a + -1) * 2a + -5 Reorder the terms: 7a + -4 + -10 + -2a + 256(-3 + a) + 2a = 4(a + -1) * 2a + -5 7a + -4 + -10 + -2a + (-3 * 256 + a * 256) + 2a = 4(a + -1) * 2a + -5 7a + -4 + -10 + -2a + (-768 + 256a) + 2a = 4(a + -1) * 2a + -5 Reorder the terms: -4 + -10 + -768 + 7a + -2a + 256a + 2a = 4(a + -1) * 2a + -5 Combine like terms: -4 + -10 = -14 -14 + -768 + 7a + -2a + 256a + 2a = 4(a + -1) * 2a + -5 Combine like terms: -14 + -768 = -782 -782 + 7a + -2a + 256a + 2a = 4(a + -1) * 2a + -5 Combine like terms: 7a + -2a = 5a -782 + 5a + 256a + 2a = 4(a + -1) * 2a + -5 Combine like terms: 5a + 256a = 261a -782 + 261a + 2a = 4(a + -1) * 2a + -5 Combine like terms: 261a + 2a = 263a -782 + 263a = 4(a + -1) * 2a + -5 Reorder the terms: -782 + 263a = 4(-1 + a) * 2a + -5 Reorder the terms for easier multiplication: -782 + 263a = 4 * 2a(-1 + a) + -5 Multiply 4 * 2 -782 + 263a = 8a(-1 + a) + -5 -782 + 263a = (-1 * 8a + a * 8a) + -5 -782 + 263a = (-8a + 8a2) + -5 Reorder the terms: -782 + 263a = -5 + -8a + 8a2 Solving -782 + 263a = -5 + -8a + 8a2 Solving for variable 'a'. Reorder the terms: -782 + 5 + 263a + 8a + -8a2 = -5 + -8a + 8a2 + 5 + 8a + -8a2 Combine like terms: -782 + 5 = -777 -777 + 263a + 8a + -8a2 = -5 + -8a + 8a2 + 5 + 8a + -8a2 Combine like terms: 263a + 8a = 271a -777 + 271a + -8a2 = -5 + -8a + 8a2 + 5 + 8a + -8a2 Reorder the terms: -777 + 271a + -8a2 = -5 + 5 + -8a + 8a + 8a2 + -8a2 Combine like terms: -5 + 5 = 0 -777 + 271a + -8a2 = 0 + -8a + 8a + 8a2 + -8a2 -777 + 271a + -8a2 = -8a + 8a + 8a2 + -8a2 Combine like terms: -8a + 8a = 0 -777 + 271a + -8a2 = 0 + 8a2 + -8a2 -777 + 271a + -8a2 = 8a2 + -8a2 Combine like terms: 8a2 + -8a2 = 0 -777 + 271a + -8a2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. 97.125 + -33.875a + a2 = 0 Move the constant term to the right: Add '-97.125' to each side of the equation. 97.125 + -33.875a + -97.125 + a2 = 0 + -97.125 Reorder the terms: 97.125 + -97.125 + -33.875a + a2 = 0 + -97.125 Combine like terms: 97.125 + -97.125 = 0.000 0.000 + -33.875a + a2 = 0 + -97.125 -33.875a + a2 = 0 + -97.125 Combine like terms: 0 + -97.125 = -97.125 -33.875a + a2 = -97.125 The a term is -33.875a. Take half its coefficient (-16.9375). Square it (286.8789063) and add it to both sides. Add '286.8789063' to each side of the equation. -33.875a + 286.8789063 + a2 = -97.125 + 286.8789063 Reorder the terms: 286.8789063 + -33.875a + a2 = -97.125 + 286.8789063 Combine like terms: -97.125 + 286.8789063 = 189.7539063 286.8789063 + -33.875a + a2 = 189.7539063 Factor a perfect square on the left side: (a + -16.9375)(a + -16.9375) = 189.7539063 Calculate the square root of the right side: 13.775119103 Break this problem into two subproblems by setting (a + -16.9375) equal to 13.775119103 and -13.775119103.Subproblem 1
a + -16.9375 = 13.775119103 Simplifying a + -16.9375 = 13.775119103 Reorder the terms: -16.9375 + a = 13.775119103 Solving -16.9375 + a = 13.775119103 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '16.9375' to each side of the equation. -16.9375 + 16.9375 + a = 13.775119103 + 16.9375 Combine like terms: -16.9375 + 16.9375 = 0.0000 0.0000 + a = 13.775119103 + 16.9375 a = 13.775119103 + 16.9375 Combine like terms: 13.775119103 + 16.9375 = 30.712619103 a = 30.712619103 Simplifying a = 30.712619103Subproblem 2
a + -16.9375 = -13.775119103 Simplifying a + -16.9375 = -13.775119103 Reorder the terms: -16.9375 + a = -13.775119103 Solving -16.9375 + a = -13.775119103 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '16.9375' to each side of the equation. -16.9375 + 16.9375 + a = -13.775119103 + 16.9375 Combine like terms: -16.9375 + 16.9375 = 0.0000 0.0000 + a = -13.775119103 + 16.9375 a = -13.775119103 + 16.9375 Combine like terms: -13.775119103 + 16.9375 = 3.162380897 a = 3.162380897 Simplifying a = 3.162380897Solution
The solution to the problem is based on the solutions from the subproblems. a = {30.712619103, 3.162380897}
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