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Simplifying 3a2 + a + -6 = 0 Reorder the terms: -6 + a + 3a2 = 0 Solving -6 + a + 3a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2 + 0.3333333333a + a2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 0.3333333333a + 2 + a2 = 0 + 2 Reorder the terms: -2 + 2 + 0.3333333333a + a2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 0.3333333333a + a2 = 0 + 2 0.3333333333a + a2 = 0 + 2 Combine like terms: 0 + 2 = 2 0.3333333333a + a2 = 2 The a term is a. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.3333333333a + 0.25 + a2 = 2 + 0.25 Reorder the terms: 0.25 + 0.3333333333a + a2 = 2 + 0.25 Combine like terms: 2 + 0.25 = 2.25 0.25 + 0.3333333333a + a2 = 2.25 Factor a perfect square on the left side: (a + 0.5)(a + 0.5) = 2.25 Calculate the square root of the right side: 1.5 Break this problem into two subproblems by setting (a + 0.5) equal to 1.5 and -1.5.Subproblem 1
a + 0.5 = 1.5 Simplifying a + 0.5 = 1.5 Reorder the terms: 0.5 + a = 1.5 Solving 0.5 + a = 1.5 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 1.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 1.5 + -0.5 a = 1.5 + -0.5 Combine like terms: 1.5 + -0.5 = 1 a = 1 Simplifying a = 1Subproblem 2
a + 0.5 = -1.5 Simplifying a + 0.5 = -1.5 Reorder the terms: 0.5 + a = -1.5 Solving 0.5 + a = -1.5 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -1.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -1.5 + -0.5 a = -1.5 + -0.5 Combine like terms: -1.5 + -0.5 = -2 a = -2 Simplifying a = -2Solution
The solution to the problem is based on the solutions from the subproblems. a = {1, -2}
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