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Simplifying a2 + 0.333a = 3 Reorder the terms: 0.333a + a2 = 3 Solving 0.333a + a2 = 3 Solving for variable 'a'. Reorder the terms: -3 + 0.333a + a2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + 0.333a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + 0.333a + 3 + a2 = 0 + 3 Reorder the terms: -3 + 3 + 0.333a + a2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 0.333a + a2 = 0 + 3 0.333a + a2 = 0 + 3 Combine like terms: 0 + 3 = 3 0.333a + a2 = 3 The a term is 0.333a. Take half its coefficient (0.1665). Square it (0.02772225) and add it to both sides. Add '0.02772225' to each side of the equation. 0.333a + 0.02772225 + a2 = 3 + 0.02772225 Reorder the terms: 0.02772225 + 0.333a + a2 = 3 + 0.02772225 Combine like terms: 3 + 0.02772225 = 3.02772225 0.02772225 + 0.333a + a2 = 3.02772225 Factor a perfect square on the left side: (a + 0.1665)(a + 0.1665) = 3.02772225 Calculate the square root of the right side: 1.740035129 Break this problem into two subproblems by setting (a + 0.1665) equal to 1.740035129 and -1.740035129.Subproblem 1
a + 0.1665 = 1.740035129 Simplifying a + 0.1665 = 1.740035129 Reorder the terms: 0.1665 + a = 1.740035129 Solving 0.1665 + a = 1.740035129 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.1665' to each side of the equation. 0.1665 + -0.1665 + a = 1.740035129 + -0.1665 Combine like terms: 0.1665 + -0.1665 = 0.0000 0.0000 + a = 1.740035129 + -0.1665 a = 1.740035129 + -0.1665 Combine like terms: 1.740035129 + -0.1665 = 1.573535129 a = 1.573535129 Simplifying a = 1.573535129Subproblem 2
a + 0.1665 = -1.740035129 Simplifying a + 0.1665 = -1.740035129 Reorder the terms: 0.1665 + a = -1.740035129 Solving 0.1665 + a = -1.740035129 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.1665' to each side of the equation. 0.1665 + -0.1665 + a = -1.740035129 + -0.1665 Combine like terms: 0.1665 + -0.1665 = 0.0000 0.0000 + a = -1.740035129 + -0.1665 a = -1.740035129 + -0.1665 Combine like terms: -1.740035129 + -0.1665 = -1.906535129 a = -1.906535129 Simplifying a = -1.906535129Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.573535129, -1.906535129}
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