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Simplifying 6a[3a + -2(3a + -5)] + 8 = 33 Reorder the terms: 6a[3a + -2(-5 + 3a)] + 8 = 33 6a[3a + (-5 * -2 + 3a * -2)] + 8 = 33 6a[3a + (10 + -6a)] + 8 = 33 Reorder the terms: 6a[10 + 3a + -6a] + 8 = 33 Combine like terms: 3a + -6a = -3a 6a[10 + -3a] + 8 = 33 [10 * 6a + -3a * 6a] + 8 = 33 [60a + -18a2] + 8 = 33 Reorder the terms: 8 + 60a + -18a2 = 33 Solving 8 + 60a + -18a2 = 33 Solving for variable 'a'. Reorder the terms: 8 + -33 + 60a + -18a2 = 33 + -33 Combine like terms: 8 + -33 = -25 -25 + 60a + -18a2 = 33 + -33 Combine like terms: 33 + -33 = 0 -25 + 60a + -18a2 = 0 Begin completing the square. Divide all terms by -18 the coefficient of the squared term: Divide each side by '-18'. 1.388888889 + -3.333333333a + a2 = 0 Move the constant term to the right: Add '-1.388888889' to each side of the equation. 1.388888889 + -3.333333333a + -1.388888889 + a2 = 0 + -1.388888889 Reorder the terms: 1.388888889 + -1.388888889 + -3.333333333a + a2 = 0 + -1.388888889 Combine like terms: 1.388888889 + -1.388888889 = 0.000000000 0.000000000 + -3.333333333a + a2 = 0 + -1.388888889 -3.333333333a + a2 = 0 + -1.388888889 Combine like terms: 0 + -1.388888889 = -1.388888889 -3.333333333a + a2 = -1.388888889 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 = -1.388888889 + 2.777777779 Reorder the terms: 2.777777779 + -3.333333333a + a2 = -1.388888889 + 2.777777779 Combine like terms: -1.388888889 + 2.777777779 = 1.38888889 2.777777779 + -3.333333333a + a2 = 1.38888889 Factor a perfect square on the left side: (a + -1.666666667)(a + -1.666666667) = 1.38888889 Calculate the square root of the right side: 1.178511302 Break this problem into two subproblems by setting (a + -1.666666667) equal to 1.178511302 and -1.178511302.Subproblem 1
a + -1.666666667 = 1.178511302 Simplifying a + -1.666666667 = 1.178511302 Reorder the terms: -1.666666667 + a = 1.178511302 Solving -1.666666667 + a = 1.178511302 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 = 1.178511302 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + a = 1.178511302 + 1.666666667 a = 1.178511302 + 1.666666667 Combine like terms: 1.178511302 + 1.666666667 = 2.845177969 a = 2.845177969 Simplifying a = 2.845177969Subproblem 2
a + -1.666666667 = -1.178511302 Simplifying a + -1.666666667 = -1.178511302 Reorder the terms: -1.666666667 + a = -1.178511302 Solving -1.666666667 + a = -1.178511302 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 = -1.178511302 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + a = -1.178511302 + 1.666666667 a = -1.178511302 + 1.666666667 Combine like terms: -1.178511302 + 1.666666667 = 0.488155365 a = 0.488155365 Simplifying a = 0.488155365Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.845177969, 0.488155365}
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