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Simplifying 3a2 + 6a + 2 = 0 Reorder the terms: 2 + 6a + 3a2 = 0 Solving 2 + 6a + 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'. 0.6666666667 + 2a + a2 = 0 Move the constant term to the right: Add '-0.6666666667' to each side of the equation. 0.6666666667 + 2a + -0.6666666667 + a2 = 0 + -0.6666666667 Reorder the terms: 0.6666666667 + -0.6666666667 + 2a + a2 = 0 + -0.6666666667 Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000 0.0000000000 + 2a + a2 = 0 + -0.6666666667 2a + a2 = 0 + -0.6666666667 Combine like terms: 0 + -0.6666666667 = -0.6666666667 2a + a2 = -0.6666666667 The a term is 2a. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2a + 1 + a2 = -0.6666666667 + 1 Reorder the terms: 1 + 2a + a2 = -0.6666666667 + 1 Combine like terms: -0.6666666667 + 1 = 0.3333333333 1 + 2a + a2 = 0.3333333333 Factor a perfect square on the left side: (a + 1)(a + 1) = 0.3333333333 Calculate the square root of the right side: 0.577350269 Break this problem into two subproblems by setting (a + 1) equal to 0.577350269 and -0.577350269.Subproblem 1
a + 1 = 0.577350269 Simplifying a + 1 = 0.577350269 Reorder the terms: 1 + a = 0.577350269 Solving 1 + a = 0.577350269 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + a = 0.577350269 + -1 Combine like terms: 1 + -1 = 0 0 + a = 0.577350269 + -1 a = 0.577350269 + -1 Combine like terms: 0.577350269 + -1 = -0.422649731 a = -0.422649731 Simplifying a = -0.422649731Subproblem 2
a + 1 = -0.577350269 Simplifying a + 1 = -0.577350269 Reorder the terms: 1 + a = -0.577350269 Solving 1 + a = -0.577350269 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + a = -0.577350269 + -1 Combine like terms: 1 + -1 = 0 0 + a = -0.577350269 + -1 a = -0.577350269 + -1 Combine like terms: -0.577350269 + -1 = -1.577350269 a = -1.577350269 Simplifying a = -1.577350269Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.422649731, -1.577350269}
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