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Simplifying 8 = a2 + 9a Reorder the terms: 8 = 9a + a2 Solving 8 = 9a + a2 Solving for variable 'a'. Reorder the terms: 8 + -9a + -1a2 = 9a + -9a + a2 + -1a2 Combine like terms: 9a + -9a = 0 8 + -9a + -1a2 = 0 + a2 + -1a2 8 + -9a + -1a2 = a2 + -1a2 Combine like terms: a2 + -1a2 = 0 8 + -9a + -1a2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -8 + 9a + a2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + 9a + 8 + a2 = 0 + 8 Reorder the terms: -8 + 8 + 9a + a2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + 9a + a2 = 0 + 8 9a + a2 = 0 + 8 Combine like terms: 0 + 8 = 8 9a + a2 = 8 The a term is 9a. Take half its coefficient (4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. 9a + 20.25 + a2 = 8 + 20.25 Reorder the terms: 20.25 + 9a + a2 = 8 + 20.25 Combine like terms: 8 + 20.25 = 28.25 20.25 + 9a + a2 = 28.25 Factor a perfect square on the left side: (a + 4.5)(a + 4.5) = 28.25 Calculate the square root of the right side: 5.315072906 Break this problem into two subproblems by setting (a + 4.5) equal to 5.315072906 and -5.315072906.Subproblem 1
a + 4.5 = 5.315072906 Simplifying a + 4.5 = 5.315072906 Reorder the terms: 4.5 + a = 5.315072906 Solving 4.5 + a = 5.315072906 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + a = 5.315072906 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + a = 5.315072906 + -4.5 a = 5.315072906 + -4.5 Combine like terms: 5.315072906 + -4.5 = 0.815072906 a = 0.815072906 Simplifying a = 0.815072906Subproblem 2
a + 4.5 = -5.315072906 Simplifying a + 4.5 = -5.315072906 Reorder the terms: 4.5 + a = -5.315072906 Solving 4.5 + a = -5.315072906 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + a = -5.315072906 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + a = -5.315072906 + -4.5 a = -5.315072906 + -4.5 Combine like terms: -5.315072906 + -4.5 = -9.815072906 a = -9.815072906 Simplifying a = -9.815072906Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.815072906, -9.815072906}
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