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Simplifying -32a2 + -44a + 15 = 0 Reorder the terms: 15 + -44a + -32a2 = 0 Solving 15 + -44a + -32a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by -32 the coefficient of the squared term: Divide each side by '-32'. -0.46875 + 1.375a + a2 = 0 Move the constant term to the right: Add '0.46875' to each side of the equation. -0.46875 + 1.375a + 0.46875 + a2 = 0 + 0.46875 Reorder the terms: -0.46875 + 0.46875 + 1.375a + a2 = 0 + 0.46875 Combine like terms: -0.46875 + 0.46875 = 0.00000 0.00000 + 1.375a + a2 = 0 + 0.46875 1.375a + a2 = 0 + 0.46875 Combine like terms: 0 + 0.46875 = 0.46875 1.375a + a2 = 0.46875 The a term is 1.375a. Take half its coefficient (0.6875). Square it (0.47265625) and add it to both sides. Add '0.47265625' to each side of the equation. 1.375a + 0.47265625 + a2 = 0.46875 + 0.47265625 Reorder the terms: 0.47265625 + 1.375a + a2 = 0.46875 + 0.47265625 Combine like terms: 0.46875 + 0.47265625 = 0.94140625 0.47265625 + 1.375a + a2 = 0.94140625 Factor a perfect square on the left side: (a + 0.6875)(a + 0.6875) = 0.94140625 Calculate the square root of the right side: 0.970260919 Break this problem into two subproblems by setting (a + 0.6875) equal to 0.970260919 and -0.970260919.Subproblem 1
a + 0.6875 = 0.970260919 Simplifying a + 0.6875 = 0.970260919 Reorder the terms: 0.6875 + a = 0.970260919 Solving 0.6875 + a = 0.970260919 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.6875' to each side of the equation. 0.6875 + -0.6875 + a = 0.970260919 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + a = 0.970260919 + -0.6875 a = 0.970260919 + -0.6875 Combine like terms: 0.970260919 + -0.6875 = 0.282760919 a = 0.282760919 Simplifying a = 0.282760919Subproblem 2
a + 0.6875 = -0.970260919 Simplifying a + 0.6875 = -0.970260919 Reorder the terms: 0.6875 + a = -0.970260919 Solving 0.6875 + a = -0.970260919 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.6875' to each side of the equation. 0.6875 + -0.6875 + a = -0.970260919 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + a = -0.970260919 + -0.6875 a = -0.970260919 + -0.6875 Combine like terms: -0.970260919 + -0.6875 = -1.657760919 a = -1.657760919 Simplifying a = -1.657760919Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.282760919, -1.657760919}
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