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Simplifying -16a2 + -5a + 312 = 0 Reorder the terms: 312 + -5a + -16a2 = 0 Solving 312 + -5a + -16a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -19.5 + 0.3125a + a2 = 0 Move the constant term to the right: Add '19.5' to each side of the equation. -19.5 + 0.3125a + 19.5 + a2 = 0 + 19.5 Reorder the terms: -19.5 + 19.5 + 0.3125a + a2 = 0 + 19.5 Combine like terms: -19.5 + 19.5 = 0.0 0.0 + 0.3125a + a2 = 0 + 19.5 0.3125a + a2 = 0 + 19.5 Combine like terms: 0 + 19.5 = 19.5 0.3125a + a2 = 19.5 The a term is 0.3125a. Take half its coefficient (0.15625). Square it (0.0244140625) and add it to both sides. Add '0.0244140625' to each side of the equation. 0.3125a + 0.0244140625 + a2 = 19.5 + 0.0244140625 Reorder the terms: 0.0244140625 + 0.3125a + a2 = 19.5 + 0.0244140625 Combine like terms: 19.5 + 0.0244140625 = 19.5244140625 0.0244140625 + 0.3125a + a2 = 19.5244140625 Factor a perfect square on the left side: (a + 0.15625)(a + 0.15625) = 19.5244140625 Calculate the square root of the right side: 4.418643917 Break this problem into two subproblems by setting (a + 0.15625) equal to 4.418643917 and -4.418643917.Subproblem 1
a + 0.15625 = 4.418643917 Simplifying a + 0.15625 = 4.418643917 Reorder the terms: 0.15625 + a = 4.418643917 Solving 0.15625 + a = 4.418643917 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.15625' to each side of the equation. 0.15625 + -0.15625 + a = 4.418643917 + -0.15625 Combine like terms: 0.15625 + -0.15625 = 0.00000 0.00000 + a = 4.418643917 + -0.15625 a = 4.418643917 + -0.15625 Combine like terms: 4.418643917 + -0.15625 = 4.262393917 a = 4.262393917 Simplifying a = 4.262393917Subproblem 2
a + 0.15625 = -4.418643917 Simplifying a + 0.15625 = -4.418643917 Reorder the terms: 0.15625 + a = -4.418643917 Solving 0.15625 + a = -4.418643917 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.15625' to each side of the equation. 0.15625 + -0.15625 + a = -4.418643917 + -0.15625 Combine like terms: 0.15625 + -0.15625 = 0.00000 0.00000 + a = -4.418643917 + -0.15625 a = -4.418643917 + -0.15625 Combine like terms: -4.418643917 + -0.15625 = -4.574893917 a = -4.574893917 Simplifying a = -4.574893917Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.262393917, -4.574893917}
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