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Simplifying a2 + 0.333333a = 3 Reorder the terms: 0.333333a + a2 = 3 Solving 0.333333a + a2 = 3 Solving for variable 'a'. Reorder the terms: -3 + 0.333333a + a2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + 0.333333a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + 0.333333a + 3 + a2 = 0 + 3 Reorder the terms: -3 + 3 + 0.333333a + a2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 0.333333a + a2 = 0 + 3 0.333333a + a2 = 0 + 3 Combine like terms: 0 + 3 = 3 0.333333a + a2 = 3 The a term is 0.333333a. Take half its coefficient (0.1666665). Square it (0.02777772222) and add it to both sides. Add '0.02777772222' to each side of the equation. 0.333333a + 0.02777772222 + a2 = 3 + 0.02777772222 Reorder the terms: 0.02777772222 + 0.333333a + a2 = 3 + 0.02777772222 Combine like terms: 3 + 0.02777772222 = 3.02777772222 0.02777772222 + 0.333333a + a2 = 3.02777772222 Factor a perfect square on the left side: (a + 0.1666665)(a + 0.1666665) = 3.02777772222 Calculate the square root of the right side: 1.740051069 Break this problem into two subproblems by setting (a + 0.1666665) equal to 1.740051069 and -1.740051069.Subproblem 1
a + 0.1666665 = 1.740051069 Simplifying a + 0.1666665 = 1.740051069 Reorder the terms: 0.1666665 + a = 1.740051069 Solving 0.1666665 + a = 1.740051069 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.1666665' to each side of the equation. 0.1666665 + -0.1666665 + a = 1.740051069 + -0.1666665 Combine like terms: 0.1666665 + -0.1666665 = 0.0000000 0.0000000 + a = 1.740051069 + -0.1666665 a = 1.740051069 + -0.1666665 Combine like terms: 1.740051069 + -0.1666665 = 1.573384569 a = 1.573384569 Simplifying a = 1.573384569Subproblem 2
a + 0.1666665 = -1.740051069 Simplifying a + 0.1666665 = -1.740051069 Reorder the terms: 0.1666665 + a = -1.740051069 Solving 0.1666665 + a = -1.740051069 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.1666665' to each side of the equation. 0.1666665 + -0.1666665 + a = -1.740051069 + -0.1666665 Combine like terms: 0.1666665 + -0.1666665 = 0.0000000 0.0000000 + a = -1.740051069 + -0.1666665 a = -1.740051069 + -0.1666665 Combine like terms: -1.740051069 + -0.1666665 = -1.906717569 a = -1.906717569 Simplifying a = -1.906717569Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.573384569, -1.906717569}
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