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Simplifying (5a2 + 8a + -9) + (a + -4) + (15a2 + -4a + 9) = 0 Reorder the terms: (-9 + 8a + 5a2) + (a + -4) + (15a2 + -4a + 9) = 0 Remove parenthesis around (-9 + 8a + 5a2) -9 + 8a + 5a2 + (a + -4) + (15a2 + -4a + 9) = 0 Reorder the terms: -9 + 8a + 5a2 + (-4 + a) + (15a2 + -4a + 9) = 0 Remove parenthesis around (-4 + a) -9 + 8a + 5a2 + -4 + a + (15a2 + -4a + 9) = 0 Reorder the terms: -9 + 8a + 5a2 + -4 + a + (9 + -4a + 15a2) = 0 Remove parenthesis around (9 + -4a + 15a2) -9 + 8a + 5a2 + -4 + a + 9 + -4a + 15a2 = 0 Reorder the terms: -9 + -4 + 9 + 8a + a + -4a + 5a2 + 15a2 = 0 Combine like terms: -9 + -4 = -13 -13 + 9 + 8a + a + -4a + 5a2 + 15a2 = 0 Combine like terms: -13 + 9 = -4 -4 + 8a + a + -4a + 5a2 + 15a2 = 0 Combine like terms: 8a + a = 9a -4 + 9a + -4a + 5a2 + 15a2 = 0 Combine like terms: 9a + -4a = 5a -4 + 5a + 5a2 + 15a2 = 0 Combine like terms: 5a2 + 15a2 = 20a2 -4 + 5a + 20a2 = 0 Solving -4 + 5a + 20a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. -0.2 + 0.25a + a2 = 0.0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + 0.25a + 0.2 + a2 = 0.0 + 0.2 Reorder the terms: -0.2 + 0.2 + 0.25a + a2 = 0.0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + 0.25a + a2 = 0.0 + 0.2 0.25a + a2 = 0.0 + 0.2 Combine like terms: 0.0 + 0.2 = 0.2 0.25a + a2 = 0.2 The a term is 0.25a. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25a + 0.015625 + a2 = 0.2 + 0.015625 Reorder the terms: 0.015625 + 0.25a + a2 = 0.2 + 0.015625 Combine like terms: 0.2 + 0.015625 = 0.215625 0.015625 + 0.25a + a2 = 0.215625 Factor a perfect square on the left side: (a + 0.125)(a + 0.125) = 0.215625 Calculate the square root of the right side: 0.464354391 Break this problem into two subproblems by setting (a + 0.125) equal to 0.464354391 and -0.464354391.Subproblem 1
a + 0.125 = 0.464354391 Simplifying a + 0.125 = 0.464354391 Reorder the terms: 0.125 + a = 0.464354391 Solving 0.125 + a = 0.464354391 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + a = 0.464354391 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + a = 0.464354391 + -0.125 a = 0.464354391 + -0.125 Combine like terms: 0.464354391 + -0.125 = 0.339354391 a = 0.339354391 Simplifying a = 0.339354391Subproblem 2
a + 0.125 = -0.464354391 Simplifying a + 0.125 = -0.464354391 Reorder the terms: 0.125 + a = -0.464354391 Solving 0.125 + a = -0.464354391 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + a = -0.464354391 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + a = -0.464354391 + -0.125 a = -0.464354391 + -0.125 Combine like terms: -0.464354391 + -0.125 = -0.589354391 a = -0.589354391 Simplifying a = -0.589354391Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.339354391, -0.589354391}
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