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Simplifying a2 + -31 + -2a = -6 + -3a2 + -1a Reorder the terms: -31 + -2a + a2 = -6 + -3a2 + -1a Reorder the terms: -31 + -2a + a2 = -6 + -1a + -3a2 Solving -31 + -2a + a2 = -6 + -1a + -3a2 Solving for variable 'a'. Reorder the terms: -31 + 6 + -2a + a + a2 + 3a2 = -6 + -1a + -3a2 + 6 + a + 3a2 Combine like terms: -31 + 6 = -25 -25 + -2a + a + a2 + 3a2 = -6 + -1a + -3a2 + 6 + a + 3a2 Combine like terms: -2a + a = -1a -25 + -1a + a2 + 3a2 = -6 + -1a + -3a2 + 6 + a + 3a2 Combine like terms: a2 + 3a2 = 4a2 -25 + -1a + 4a2 = -6 + -1a + -3a2 + 6 + a + 3a2 Reorder the terms: -25 + -1a + 4a2 = -6 + 6 + -1a + a + -3a2 + 3a2 Combine like terms: -6 + 6 = 0 -25 + -1a + 4a2 = 0 + -1a + a + -3a2 + 3a2 -25 + -1a + 4a2 = -1a + a + -3a2 + 3a2 Combine like terms: -1a + a = 0 -25 + -1a + 4a2 = 0 + -3a2 + 3a2 -25 + -1a + 4a2 = -3a2 + 3a2 Combine like terms: -3a2 + 3a2 = 0 -25 + -1a + 4a2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -6.25 + -0.25a + a2 = 0 Move the constant term to the right: Add '6.25' to each side of the equation. -6.25 + -0.25a + 6.25 + a2 = 0 + 6.25 Reorder the terms: -6.25 + 6.25 + -0.25a + a2 = 0 + 6.25 Combine like terms: -6.25 + 6.25 = 0.00 0.00 + -0.25a + a2 = 0 + 6.25 -0.25a + a2 = 0 + 6.25 Combine like terms: 0 + 6.25 = 6.25 -0.25a + a2 = 6.25 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 = 6.25 + 0.015625 Reorder the terms: 0.015625 + -0.25a + a2 = 6.25 + 0.015625 Combine like terms: 6.25 + 0.015625 = 6.265625 0.015625 + -0.25a + a2 = 6.265625 Factor a perfect square on the left side: (a + -0.125)(a + -0.125) = 6.265625 Calculate the square root of the right side: 2.503123049 Break this problem into two subproblems by setting (a + -0.125) equal to 2.503123049 and -2.503123049.Subproblem 1
a + -0.125 = 2.503123049 Simplifying a + -0.125 = 2.503123049 Reorder the terms: -0.125 + a = 2.503123049 Solving -0.125 + a = 2.503123049 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 = 2.503123049 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + a = 2.503123049 + 0.125 a = 2.503123049 + 0.125 Combine like terms: 2.503123049 + 0.125 = 2.628123049 a = 2.628123049 Simplifying a = 2.628123049Subproblem 2
a + -0.125 = -2.503123049 Simplifying a + -0.125 = -2.503123049 Reorder the terms: -0.125 + a = -2.503123049 Solving -0.125 + a = -2.503123049 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 = -2.503123049 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + a = -2.503123049 + 0.125 a = -2.503123049 + 0.125 Combine like terms: -2.503123049 + 0.125 = -2.378123049 a = -2.378123049 Simplifying a = -2.378123049Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.628123049, -2.378123049}
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