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Simplifying 0 = a2 + -12a + 33 Reorder the terms: 0 = 33 + -12a + a2 Solving 0 = 33 + -12a + a2 Solving for variable 'a'. Combine like terms: 0 + -33 = -33 -33 + 12a + -1a2 = 33 + -12a + a2 + -33 + 12a + -1a2 Reorder the terms: -33 + 12a + -1a2 = 33 + -33 + -12a + 12a + a2 + -1a2 Combine like terms: 33 + -33 = 0 -33 + 12a + -1a2 = 0 + -12a + 12a + a2 + -1a2 -33 + 12a + -1a2 = -12a + 12a + a2 + -1a2 Combine like terms: -12a + 12a = 0 -33 + 12a + -1a2 = 0 + a2 + -1a2 -33 + 12a + -1a2 = a2 + -1a2 Combine like terms: a2 + -1a2 = 0 -33 + 12a + -1a2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 33 + -12a + a2 = 0 Move the constant term to the right: Add '-33' to each side of the equation. 33 + -12a + -33 + a2 = 0 + -33 Reorder the terms: 33 + -33 + -12a + a2 = 0 + -33 Combine like terms: 33 + -33 = 0 0 + -12a + a2 = 0 + -33 -12a + a2 = 0 + -33 Combine like terms: 0 + -33 = -33 -12a + a2 = -33 The a term is -12a. Take half its coefficient (-6). Square it (36) and add it to both sides. Add '36' to each side of the equation. -12a + 36 + a2 = -33 + 36 Reorder the terms: 36 + -12a + a2 = -33 + 36 Combine like terms: -33 + 36 = 3 36 + -12a + a2 = 3 Factor a perfect square on the left side: (a + -6)(a + -6) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (a + -6) equal to 1.732050808 and -1.732050808.Subproblem 1
a + -6 = 1.732050808 Simplifying a + -6 = 1.732050808 Reorder the terms: -6 + a = 1.732050808 Solving -6 + a = 1.732050808 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + a = 1.732050808 + 6 Combine like terms: -6 + 6 = 0 0 + a = 1.732050808 + 6 a = 1.732050808 + 6 Combine like terms: 1.732050808 + 6 = 7.732050808 a = 7.732050808 Simplifying a = 7.732050808Subproblem 2
a + -6 = -1.732050808 Simplifying a + -6 = -1.732050808 Reorder the terms: -6 + a = -1.732050808 Solving -6 + a = -1.732050808 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + a = -1.732050808 + 6 Combine like terms: -6 + 6 = 0 0 + a = -1.732050808 + 6 a = -1.732050808 + 6 Combine like terms: -1.732050808 + 6 = 4.267949192 a = 4.267949192 Simplifying a = 4.267949192Solution
The solution to the problem is based on the solutions from the subproblems. a = {7.732050808, 4.267949192}
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