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Simplifying 8a2 + 28a + -31 = 0 Reorder the terms: -31 + 28a + 8a2 = 0 Solving -31 + 28a + 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 + 3.5a + a2 = 0 Move the constant term to the right: Add '3.875' to each side of the equation. -3.875 + 3.5a + 3.875 + a2 = 0 + 3.875 Reorder the terms: -3.875 + 3.875 + 3.5a + a2 = 0 + 3.875 Combine like terms: -3.875 + 3.875 = 0.000 0.000 + 3.5a + a2 = 0 + 3.875 3.5a + a2 = 0 + 3.875 Combine like terms: 0 + 3.875 = 3.875 3.5a + a2 = 3.875 The a term is 3.5a. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5a + 3.0625 + a2 = 3.875 + 3.0625 Reorder the terms: 3.0625 + 3.5a + a2 = 3.875 + 3.0625 Combine like terms: 3.875 + 3.0625 = 6.9375 3.0625 + 3.5a + a2 = 6.9375 Factor a perfect square on the left side: (a + 1.75)(a + 1.75) = 6.9375 Calculate the square root of the right side: 2.633913438 Break this problem into two subproblems by setting (a + 1.75) equal to 2.633913438 and -2.633913438.Subproblem 1
a + 1.75 = 2.633913438 Simplifying a + 1.75 = 2.633913438 Reorder the terms: 1.75 + a = 2.633913438 Solving 1.75 + a = 2.633913438 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + a = 2.633913438 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + a = 2.633913438 + -1.75 a = 2.633913438 + -1.75 Combine like terms: 2.633913438 + -1.75 = 0.883913438 a = 0.883913438 Simplifying a = 0.883913438Subproblem 2
a + 1.75 = -2.633913438 Simplifying a + 1.75 = -2.633913438 Reorder the terms: 1.75 + a = -2.633913438 Solving 1.75 + a = -2.633913438 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + a = -2.633913438 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + a = -2.633913438 + -1.75 a = -2.633913438 + -1.75 Combine like terms: -2.633913438 + -1.75 = -4.383913438 a = -4.383913438 Simplifying a = -4.383913438Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.883913438, -4.383913438}
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