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Simplifying a2 + 31a + 40 = 0 Reorder the terms: 40 + 31a + a2 = 0 Solving 40 + 31a + a2 = 0 Solving for variable 'a'. Begin completing the square. Move the constant term to the right: Add '-40' to each side of the equation. 40 + 31a + -40 + a2 = 0 + -40 Reorder the terms: 40 + -40 + 31a + a2 = 0 + -40 Combine like terms: 40 + -40 = 0 0 + 31a + a2 = 0 + -40 31a + a2 = 0 + -40 Combine like terms: 0 + -40 = -40 31a + a2 = -40 The a term is 31a. Take half its coefficient (15.5). Square it (240.25) and add it to both sides. Add '240.25' to each side of the equation. 31a + 240.25 + a2 = -40 + 240.25 Reorder the terms: 240.25 + 31a + a2 = -40 + 240.25 Combine like terms: -40 + 240.25 = 200.25 240.25 + 31a + a2 = 200.25 Factor a perfect square on the left side: (a + 15.5)(a + 15.5) = 200.25 Calculate the square root of the right side: 14.150971698 Break this problem into two subproblems by setting (a + 15.5) equal to 14.150971698 and -14.150971698.Subproblem 1
a + 15.5 = 14.150971698 Simplifying a + 15.5 = 14.150971698 Reorder the terms: 15.5 + a = 14.150971698 Solving 15.5 + a = 14.150971698 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-15.5' to each side of the equation. 15.5 + -15.5 + a = 14.150971698 + -15.5 Combine like terms: 15.5 + -15.5 = 0.0 0.0 + a = 14.150971698 + -15.5 a = 14.150971698 + -15.5 Combine like terms: 14.150971698 + -15.5 = -1.349028302 a = -1.349028302 Simplifying a = -1.349028302Subproblem 2
a + 15.5 = -14.150971698 Simplifying a + 15.5 = -14.150971698 Reorder the terms: 15.5 + a = -14.150971698 Solving 15.5 + a = -14.150971698 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-15.5' to each side of the equation. 15.5 + -15.5 + a = -14.150971698 + -15.5 Combine like terms: 15.5 + -15.5 = 0.0 0.0 + a = -14.150971698 + -15.5 a = -14.150971698 + -15.5 Combine like terms: -14.150971698 + -15.5 = -29.650971698 a = -29.650971698 Simplifying a = -29.650971698Solution
The solution to the problem is based on the solutions from the subproblems. a = {-1.349028302, -29.650971698}
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