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Simplifying 8a2 + -4a + -31 = 0 Reorder the terms: -31 + -4a + 8a2 = 0 Solving -31 + -4a + 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'. -3.875 + -0.5a + a2 = 0 Move the constant term to the right: Add '3.875' to each side of the equation. -3.875 + -0.5a + 3.875 + a2 = 0 + 3.875 Reorder the terms: -3.875 + 3.875 + -0.5a + a2 = 0 + 3.875 Combine like terms: -3.875 + 3.875 = 0.000 0.000 + -0.5a + a2 = 0 + 3.875 -0.5a + a2 = 0 + 3.875 Combine like terms: 0 + 3.875 = 3.875 -0.5a + a2 = 3.875 The a term is -0.5a. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5a + 0.0625 + a2 = 3.875 + 0.0625 Reorder the terms: 0.0625 + -0.5a + a2 = 3.875 + 0.0625 Combine like terms: 3.875 + 0.0625 = 3.9375 0.0625 + -0.5a + a2 = 3.9375 Factor a perfect square on the left side: (a + -0.25)(a + -0.25) = 3.9375 Calculate the square root of the right side: 1.984313483 Break this problem into two subproblems by setting (a + -0.25) equal to 1.984313483 and -1.984313483.Subproblem 1
a + -0.25 = 1.984313483 Simplifying a + -0.25 = 1.984313483 Reorder the terms: -0.25 + a = 1.984313483 Solving -0.25 + a = 1.984313483 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + a = 1.984313483 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = 1.984313483 + 0.25 a = 1.984313483 + 0.25 Combine like terms: 1.984313483 + 0.25 = 2.234313483 a = 2.234313483 Simplifying a = 2.234313483Subproblem 2
a + -0.25 = -1.984313483 Simplifying a + -0.25 = -1.984313483 Reorder the terms: -0.25 + a = -1.984313483 Solving -0.25 + a = -1.984313483 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + a = -1.984313483 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = -1.984313483 + 0.25 a = -1.984313483 + 0.25 Combine like terms: -1.984313483 + 0.25 = -1.734313483 a = -1.734313483 Simplifying a = -1.734313483Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.234313483, -1.734313483}
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