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Simplifying 8a * a + 37a + -15 = a Multiply a * a 8a2 + 37a + -15 = a Reorder the terms: -15 + 37a + 8a2 = a Solving -15 + 37a + 8a2 = a Solving for variable 'a'. Reorder the terms: -15 + 37a + -1a + 8a2 = a + -1a Combine like terms: 37a + -1a = 36a -15 + 36a + 8a2 = a + -1a Combine like terms: a + -1a = 0 -15 + 36a + 8a2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -1.875 + 4.5a + a2 = 0 Move the constant term to the right: Add '1.875' to each side of the equation. -1.875 + 4.5a + 1.875 + a2 = 0 + 1.875 Reorder the terms: -1.875 + 1.875 + 4.5a + a2 = 0 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + 4.5a + a2 = 0 + 1.875 4.5a + a2 = 0 + 1.875 Combine like terms: 0 + 1.875 = 1.875 4.5a + a2 = 1.875 The a term is 4.5a. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5a + 5.0625 + a2 = 1.875 + 5.0625 Reorder the terms: 5.0625 + 4.5a + a2 = 1.875 + 5.0625 Combine like terms: 1.875 + 5.0625 = 6.9375 5.0625 + 4.5a + a2 = 6.9375 Factor a perfect square on the left side: (a + 2.25)(a + 2.25) = 6.9375 Calculate the square root of the right side: 2.633913438 Break this problem into two subproblems by setting (a + 2.25) equal to 2.633913438 and -2.633913438.Subproblem 1
a + 2.25 = 2.633913438 Simplifying a + 2.25 = 2.633913438 Reorder the terms: 2.25 + a = 2.633913438 Solving 2.25 + a = 2.633913438 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + a = 2.633913438 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + a = 2.633913438 + -2.25 a = 2.633913438 + -2.25 Combine like terms: 2.633913438 + -2.25 = 0.383913438 a = 0.383913438 Simplifying a = 0.383913438Subproblem 2
a + 2.25 = -2.633913438 Simplifying a + 2.25 = -2.633913438 Reorder the terms: 2.25 + a = -2.633913438 Solving 2.25 + a = -2.633913438 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + a = -2.633913438 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + a = -2.633913438 + -2.25 a = -2.633913438 + -2.25 Combine like terms: -2.633913438 + -2.25 = -4.883913438 a = -4.883913438 Simplifying a = -4.883913438Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.383913438, -4.883913438}
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