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Simplifying 1a(3a + 1) + (a + 1)(a + -1) = 0 Reorder the terms: 1a(1 + 3a) + (a + 1)(a + -1) = 0 (1 * 1a + 3a * 1a) + (a + 1)(a + -1) = 0 (1a + 3a2) + (a + 1)(a + -1) = 0 Reorder the terms: 1a + 3a2 + (1 + a)(a + -1) = 0 Reorder the terms: 1a + 3a2 + (1 + a)(-1 + a) = 0 Multiply (1 + a) * (-1 + a) 1a + 3a2 + (1(-1 + a) + a(-1 + a)) = 0 1a + 3a2 + ((-1 * 1 + a * 1) + a(-1 + a)) = 0 1a + 3a2 + ((-1 + 1a) + a(-1 + a)) = 0 1a + 3a2 + (-1 + 1a + (-1 * a + a * a)) = 0 1a + 3a2 + (-1 + 1a + (-1a + a2)) = 0 Combine like terms: 1a + -1a = 0 1a + 3a2 + (-1 + 0 + a2) = 0 1a + 3a2 + (-1 + a2) = 0 Reorder the terms: -1 + 1a + 3a2 + a2 = 0 Combine like terms: 3a2 + a2 = 4a2 -1 + 1a + 4a2 = 0 Solving -1 + 1a + 4a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + 0.25a + a2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + 0.25a + 0.25 + a2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + 0.25a + a2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + 0.25a + a2 = 0 + 0.25 0.25a + a2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 0.25a + a2 = 0.25 The a term is 0.25a. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25a + 0.015625 + a2 = 0.25 + 0.015625 Reorder the terms: 0.015625 + 0.25a + a2 = 0.25 + 0.015625 Combine like terms: 0.25 + 0.015625 = 0.265625 0.015625 + 0.25a + a2 = 0.265625 Factor a perfect square on the left side: (a + 0.125)(a + 0.125) = 0.265625 Calculate the square root of the right side: 0.515388203 Break this problem into two subproblems by setting (a + 0.125) equal to 0.515388203 and -0.515388203.Subproblem 1
a + 0.125 = 0.515388203 Simplifying a + 0.125 = 0.515388203 Reorder the terms: 0.125 + a = 0.515388203 Solving 0.125 + a = 0.515388203 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + a = 0.515388203 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + a = 0.515388203 + -0.125 a = 0.515388203 + -0.125 Combine like terms: 0.515388203 + -0.125 = 0.390388203 a = 0.390388203 Simplifying a = 0.390388203Subproblem 2
a + 0.125 = -0.515388203 Simplifying a + 0.125 = -0.515388203 Reorder the terms: 0.125 + a = -0.515388203 Solving 0.125 + a = -0.515388203 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + a = -0.515388203 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + a = -0.515388203 + -0.125 a = -0.515388203 + -0.125 Combine like terms: -0.515388203 + -0.125 = -0.640388203 a = -0.640388203 Simplifying a = -0.640388203Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.390388203, -0.640388203}
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