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Simplifying 8a2 + 3a + -15 = 0 Reorder the terms: -15 + 3a + 8a2 = 0 Solving -15 + 3a + 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'. -1.875 + 0.375a + a2 = 0 Move the constant term to the right: Add '1.875' to each side of the equation. -1.875 + 0.375a + 1.875 + a2 = 0 + 1.875 Reorder the terms: -1.875 + 1.875 + 0.375a + a2 = 0 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + 0.375a + a2 = 0 + 1.875 0.375a + a2 = 0 + 1.875 Combine like terms: 0 + 1.875 = 1.875 0.375a + a2 = 1.875 The a term is 0.375a. Take half its coefficient (0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. 0.375a + 0.03515625 + a2 = 1.875 + 0.03515625 Reorder the terms: 0.03515625 + 0.375a + a2 = 1.875 + 0.03515625 Combine like terms: 1.875 + 0.03515625 = 1.91015625 0.03515625 + 0.375a + a2 = 1.91015625 Factor a perfect square on the left side: (a + 0.1875)(a + 0.1875) = 1.91015625 Calculate the square root of the right side: 1.382084024 Break this problem into two subproblems by setting (a + 0.1875) equal to 1.382084024 and -1.382084024.Subproblem 1
a + 0.1875 = 1.382084024 Simplifying a + 0.1875 = 1.382084024 Reorder the terms: 0.1875 + a = 1.382084024 Solving 0.1875 + a = 1.382084024 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + a = 1.382084024 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + a = 1.382084024 + -0.1875 a = 1.382084024 + -0.1875 Combine like terms: 1.382084024 + -0.1875 = 1.194584024 a = 1.194584024 Simplifying a = 1.194584024Subproblem 2
a + 0.1875 = -1.382084024 Simplifying a + 0.1875 = -1.382084024 Reorder the terms: 0.1875 + a = -1.382084024 Solving 0.1875 + a = -1.382084024 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + a = -1.382084024 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + a = -1.382084024 + -0.1875 a = -1.382084024 + -0.1875 Combine like terms: -1.382084024 + -0.1875 = -1.569584024 a = -1.569584024 Simplifying a = -1.569584024Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.194584024, -1.569584024}
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