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Simplifying 3a2 + 13a + 6 = 0 Reorder the terms: 6 + 13a + 3a2 = 0 Solving 6 + 13a + 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 + 4.333333333a + a2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + 4.333333333a + -2 + a2 = 0 + -2 Reorder the terms: 2 + -2 + 4.333333333a + a2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + 4.333333333a + a2 = 0 + -2 4.333333333a + a2 = 0 + -2 Combine like terms: 0 + -2 = -2 4.333333333a + a2 = -2 The a term is 4.333333333a. Take half its coefficient (2.166666667). Square it (4.694444446) and add it to both sides. Add '4.694444446' to each side of the equation. 4.333333333a + 4.694444446 + a2 = -2 + 4.694444446 Reorder the terms: 4.694444446 + 4.333333333a + a2 = -2 + 4.694444446 Combine like terms: -2 + 4.694444446 = 2.694444446 4.694444446 + 4.333333333a + a2 = 2.694444446 Factor a perfect square on the left side: (a + 2.166666667)(a + 2.166666667) = 2.694444446 Calculate the square root of the right side: 1.641476301 Break this problem into two subproblems by setting (a + 2.166666667) equal to 1.641476301 and -1.641476301.Subproblem 1
a + 2.166666667 = 1.641476301 Simplifying a + 2.166666667 = 1.641476301 Reorder the terms: 2.166666667 + a = 1.641476301 Solving 2.166666667 + a = 1.641476301 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.166666667' to each side of the equation. 2.166666667 + -2.166666667 + a = 1.641476301 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + a = 1.641476301 + -2.166666667 a = 1.641476301 + -2.166666667 Combine like terms: 1.641476301 + -2.166666667 = -0.525190366 a = -0.525190366 Simplifying a = -0.525190366Subproblem 2
a + 2.166666667 = -1.641476301 Simplifying a + 2.166666667 = -1.641476301 Reorder the terms: 2.166666667 + a = -1.641476301 Solving 2.166666667 + a = -1.641476301 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.166666667' to each side of the equation. 2.166666667 + -2.166666667 + a = -1.641476301 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + a = -1.641476301 + -2.166666667 a = -1.641476301 + -2.166666667 Combine like terms: -1.641476301 + -2.166666667 = -3.808142968 a = -3.808142968 Simplifying a = -3.808142968Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.525190366, -3.808142968}
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