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Simplifying 9a2 + -29a + 3 = 0 Reorder the terms: 3 + -29a + 9a2 = 0 Solving 3 + -29a + 9a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 0.3333333333 + -3.222222222a + a2 = 0 Move the constant term to the right: Add '-0.3333333333' to each side of the equation. 0.3333333333 + -3.222222222a + -0.3333333333 + a2 = 0 + -0.3333333333 Reorder the terms: 0.3333333333 + -0.3333333333 + -3.222222222a + a2 = 0 + -0.3333333333 Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000 0.0000000000 + -3.222222222a + a2 = 0 + -0.3333333333 -3.222222222a + a2 = 0 + -0.3333333333 Combine like terms: 0 + -0.3333333333 = -0.3333333333 -3.222222222a + a2 = -0.3333333333 The a term is -3.222222222a. Take half its coefficient (-1.611111111). Square it (2.595679012) and add it to both sides. Add '2.595679012' to each side of the equation. -3.222222222a + 2.595679012 + a2 = -0.3333333333 + 2.595679012 Reorder the terms: 2.595679012 + -3.222222222a + a2 = -0.3333333333 + 2.595679012 Combine like terms: -0.3333333333 + 2.595679012 = 2.2623456787 2.595679012 + -3.222222222a + a2 = 2.2623456787 Factor a perfect square on the left side: (a + -1.611111111)(a + -1.611111111) = 2.2623456787 Calculate the square root of the right side: 1.504109597 Break this problem into two subproblems by setting (a + -1.611111111) equal to 1.504109597 and -1.504109597.Subproblem 1
a + -1.611111111 = 1.504109597 Simplifying a + -1.611111111 = 1.504109597 Reorder the terms: -1.611111111 + a = 1.504109597 Solving -1.611111111 + a = 1.504109597 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.611111111' to each side of the equation. -1.611111111 + 1.611111111 + a = 1.504109597 + 1.611111111 Combine like terms: -1.611111111 + 1.611111111 = 0.000000000 0.000000000 + a = 1.504109597 + 1.611111111 a = 1.504109597 + 1.611111111 Combine like terms: 1.504109597 + 1.611111111 = 3.115220708 a = 3.115220708 Simplifying a = 3.115220708Subproblem 2
a + -1.611111111 = -1.504109597 Simplifying a + -1.611111111 = -1.504109597 Reorder the terms: -1.611111111 + a = -1.504109597 Solving -1.611111111 + a = -1.504109597 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.611111111' to each side of the equation. -1.611111111 + 1.611111111 + a = -1.504109597 + 1.611111111 Combine like terms: -1.611111111 + 1.611111111 = 0.000000000 0.000000000 + a = -1.504109597 + 1.611111111 a = -1.504109597 + 1.611111111 Combine like terms: -1.504109597 + 1.611111111 = 0.107001514 a = 0.107001514 Simplifying a = 0.107001514Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.115220708, 0.107001514}
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