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Simplifying 5a2 + 8a + -31 = 0 Reorder the terms: -31 + 8a + 5a2 = 0 Solving -31 + 8a + 5a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -6.2 + 1.6a + a2 = 0 Move the constant term to the right: Add '6.2' to each side of the equation. -6.2 + 1.6a + 6.2 + a2 = 0 + 6.2 Reorder the terms: -6.2 + 6.2 + 1.6a + a2 = 0 + 6.2 Combine like terms: -6.2 + 6.2 = 0.0 0.0 + 1.6a + a2 = 0 + 6.2 1.6a + a2 = 0 + 6.2 Combine like terms: 0 + 6.2 = 6.2 1.6a + a2 = 6.2 The a term is 1.6a. Take half its coefficient (0.8). Square it (0.64) and add it to both sides. Add '0.64' to each side of the equation. 1.6a + 0.64 + a2 = 6.2 + 0.64 Reorder the terms: 0.64 + 1.6a + a2 = 6.2 + 0.64 Combine like terms: 6.2 + 0.64 = 6.84 0.64 + 1.6a + a2 = 6.84 Factor a perfect square on the left side: (a + 0.8)(a + 0.8) = 6.84 Calculate the square root of the right side: 2.615339366 Break this problem into two subproblems by setting (a + 0.8) equal to 2.615339366 and -2.615339366.Subproblem 1
a + 0.8 = 2.615339366 Simplifying a + 0.8 = 2.615339366 Reorder the terms: 0.8 + a = 2.615339366 Solving 0.8 + a = 2.615339366 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + a = 2.615339366 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + a = 2.615339366 + -0.8 a = 2.615339366 + -0.8 Combine like terms: 2.615339366 + -0.8 = 1.815339366 a = 1.815339366 Simplifying a = 1.815339366Subproblem 2
a + 0.8 = -2.615339366 Simplifying a + 0.8 = -2.615339366 Reorder the terms: 0.8 + a = -2.615339366 Solving 0.8 + a = -2.615339366 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + a = -2.615339366 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + a = -2.615339366 + -0.8 a = -2.615339366 + -0.8 Combine like terms: -2.615339366 + -0.8 = -3.415339366 a = -3.415339366 Simplifying a = -3.415339366Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.815339366, -3.415339366}
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