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Simplifying 3a2 + -10a + -12 = 0 Reorder the terms: -12 + -10a + 3a2 = 0 Solving -12 + -10a + 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'. -4 + -3.333333333a + a2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + -3.333333333a + 4 + a2 = 0 + 4 Reorder the terms: -4 + 4 + -3.333333333a + a2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -3.333333333a + a2 = 0 + 4 -3.333333333a + a2 = 0 + 4 Combine like terms: 0 + 4 = 4 -3.333333333a + a2 = 4 The a term is -3.333333333a. Take half its coefficient (-1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. -3.333333333a + 2.777777779 + a2 = 4 + 2.777777779 Reorder the terms: 2.777777779 + -3.333333333a + a2 = 4 + 2.777777779 Combine like terms: 4 + 2.777777779 = 6.777777779 2.777777779 + -3.333333333a + a2 = 6.777777779 Factor a perfect square on the left side: (a + -1.666666667)(a + -1.666666667) = 6.777777779 Calculate the square root of the right side: 2.603416559 Break this problem into two subproblems by setting (a + -1.666666667) equal to 2.603416559 and -2.603416559.Subproblem 1
a + -1.666666667 = 2.603416559 Simplifying a + -1.666666667 = 2.603416559 Reorder the terms: -1.666666667 + a = 2.603416559 Solving -1.666666667 + a = 2.603416559 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + a = 2.603416559 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + a = 2.603416559 + 1.666666667 a = 2.603416559 + 1.666666667 Combine like terms: 2.603416559 + 1.666666667 = 4.270083226 a = 4.270083226 Simplifying a = 4.270083226Subproblem 2
a + -1.666666667 = -2.603416559 Simplifying a + -1.666666667 = -2.603416559 Reorder the terms: -1.666666667 + a = -2.603416559 Solving -1.666666667 + a = -2.603416559 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + a = -2.603416559 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + a = -2.603416559 + 1.666666667 a = -2.603416559 + 1.666666667 Combine like terms: -2.603416559 + 1.666666667 = -0.936749892 a = -0.936749892 Simplifying a = -0.936749892Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.270083226, -0.936749892}
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