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Simplifying 2 * 3a2 + 3a + 1 = 60 Multiply 2 * 3 6a2 + 3a + 1 = 60 Reorder the terms: 1 + 3a + 6a2 = 60 Solving 1 + 3a + 6a2 = 60 Solving for variable 'a'. Reorder the terms: 1 + -60 + 3a + 6a2 = 60 + -60 Combine like terms: 1 + -60 = -59 -59 + 3a + 6a2 = 60 + -60 Combine like terms: 60 + -60 = 0 -59 + 3a + 6a2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -9.833333333 + 0.5a + a2 = 0 Move the constant term to the right: Add '9.833333333' to each side of the equation. -9.833333333 + 0.5a + 9.833333333 + a2 = 0 + 9.833333333 Reorder the terms: -9.833333333 + 9.833333333 + 0.5a + a2 = 0 + 9.833333333 Combine like terms: -9.833333333 + 9.833333333 = 0.000000000 0.000000000 + 0.5a + a2 = 0 + 9.833333333 0.5a + a2 = 0 + 9.833333333 Combine like terms: 0 + 9.833333333 = 9.833333333 0.5a + a2 = 9.833333333 The a term is 0.5a. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5a + 0.0625 + a2 = 9.833333333 + 0.0625 Reorder the terms: 0.0625 + 0.5a + a2 = 9.833333333 + 0.0625 Combine like terms: 9.833333333 + 0.0625 = 9.895833333 0.0625 + 0.5a + a2 = 9.895833333 Factor a perfect square on the left side: (a + 0.25)(a + 0.25) = 9.895833333 Calculate the square root of the right side: 3.145764348 Break this problem into two subproblems by setting (a + 0.25) equal to 3.145764348 and -3.145764348.Subproblem 1
a + 0.25 = 3.145764348 Simplifying a + 0.25 = 3.145764348 Reorder the terms: 0.25 + a = 3.145764348 Solving 0.25 + a = 3.145764348 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + a = 3.145764348 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + a = 3.145764348 + -0.25 a = 3.145764348 + -0.25 Combine like terms: 3.145764348 + -0.25 = 2.895764348 a = 2.895764348 Simplifying a = 2.895764348Subproblem 2
a + 0.25 = -3.145764348 Simplifying a + 0.25 = -3.145764348 Reorder the terms: 0.25 + a = -3.145764348 Solving 0.25 + a = -3.145764348 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + a = -3.145764348 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + a = -3.145764348 + -0.25 a = -3.145764348 + -0.25 Combine like terms: -3.145764348 + -0.25 = -3.395764348 a = -3.395764348 Simplifying a = -3.395764348Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.895764348, -3.395764348}
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