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Simplifying 6a2 + 9a + -5 = 0 Reorder the terms: -5 + 9a + 6a2 = 0 Solving -5 + 9a + 6a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.8333333333 + 1.5a + a2 = 0 Move the constant term to the right: Add '0.8333333333' to each side of the equation. -0.8333333333 + 1.5a + 0.8333333333 + a2 = 0 + 0.8333333333 Reorder the terms: -0.8333333333 + 0.8333333333 + 1.5a + a2 = 0 + 0.8333333333 Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000 0.0000000000 + 1.5a + a2 = 0 + 0.8333333333 1.5a + a2 = 0 + 0.8333333333 Combine like terms: 0 + 0.8333333333 = 0.8333333333 1.5a + a2 = 0.8333333333 The a term is 1.5a. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5a + 0.5625 + a2 = 0.8333333333 + 0.5625 Reorder the terms: 0.5625 + 1.5a + a2 = 0.8333333333 + 0.5625 Combine like terms: 0.8333333333 + 0.5625 = 1.3958333333 0.5625 + 1.5a + a2 = 1.3958333333 Factor a perfect square on the left side: (a + 0.75)(a + 0.75) = 1.3958333333 Calculate the square root of the right side: 1.181453907 Break this problem into two subproblems by setting (a + 0.75) equal to 1.181453907 and -1.181453907.Subproblem 1
a + 0.75 = 1.181453907 Simplifying a + 0.75 = 1.181453907 Reorder the terms: 0.75 + a = 1.181453907 Solving 0.75 + a = 1.181453907 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = 1.181453907 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = 1.181453907 + -0.75 a = 1.181453907 + -0.75 Combine like terms: 1.181453907 + -0.75 = 0.431453907 a = 0.431453907 Simplifying a = 0.431453907Subproblem 2
a + 0.75 = -1.181453907 Simplifying a + 0.75 = -1.181453907 Reorder the terms: 0.75 + a = -1.181453907 Solving 0.75 + a = -1.181453907 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = -1.181453907 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = -1.181453907 + -0.75 a = -1.181453907 + -0.75 Combine like terms: -1.181453907 + -0.75 = -1.931453907 a = -1.931453907 Simplifying a = -1.931453907Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.431453907, -1.931453907}
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