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Simplifying a2 + 6a2 + -12 + 5a + a2 = 0 Reorder the terms: -12 + 5a + a2 + 6a2 + a2 = 0 Combine like terms: a2 + 6a2 = 7a2 -12 + 5a + 7a2 + a2 = 0 Combine like terms: 7a2 + a2 = 8a2 -12 + 5a + 8a2 = 0 Solving -12 + 5a + 8a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -1.5 + 0.625a + a2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 0.625a + 1.5 + a2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 0.625a + a2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 0.625a + a2 = 0 + 1.5 0.625a + a2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 0.625a + a2 = 1.5 The a term is 0.625a. Take half its coefficient (0.3125). Square it (0.09765625) and add it to both sides. Add '0.09765625' to each side of the equation. 0.625a + 0.09765625 + a2 = 1.5 + 0.09765625 Reorder the terms: 0.09765625 + 0.625a + a2 = 1.5 + 0.09765625 Combine like terms: 1.5 + 0.09765625 = 1.59765625 0.09765625 + 0.625a + a2 = 1.59765625 Factor a perfect square on the left side: (a + 0.3125)(a + 0.3125) = 1.59765625 Calculate the square root of the right side: 1.263984276 Break this problem into two subproblems by setting (a + 0.3125) equal to 1.263984276 and -1.263984276.Subproblem 1
a + 0.3125 = 1.263984276 Simplifying a + 0.3125 = 1.263984276 Reorder the terms: 0.3125 + a = 1.263984276 Solving 0.3125 + a = 1.263984276 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.3125' to each side of the equation. 0.3125 + -0.3125 + a = 1.263984276 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + a = 1.263984276 + -0.3125 a = 1.263984276 + -0.3125 Combine like terms: 1.263984276 + -0.3125 = 0.951484276 a = 0.951484276 Simplifying a = 0.951484276Subproblem 2
a + 0.3125 = -1.263984276 Simplifying a + 0.3125 = -1.263984276 Reorder the terms: 0.3125 + a = -1.263984276 Solving 0.3125 + a = -1.263984276 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.3125' to each side of the equation. 0.3125 + -0.3125 + a = -1.263984276 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + a = -1.263984276 + -0.3125 a = -1.263984276 + -0.3125 Combine like terms: -1.263984276 + -0.3125 = -1.576484276 a = -1.576484276 Simplifying a = -1.576484276Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.951484276, -1.576484276}
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