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Simplifying 2a = 4a2 + -1 Reorder the terms: 2a = -1 + 4a2 Solving 2a = -1 + 4a2 Solving for variable 'a'. Reorder the terms: 1 + 2a + -4a2 = -1 + 4a2 + 1 + -4a2 Reorder the terms: 1 + 2a + -4a2 = -1 + 1 + 4a2 + -4a2 Combine like terms: -1 + 1 = 0 1 + 2a + -4a2 = 0 + 4a2 + -4a2 1 + 2a + -4a2 = 4a2 + -4a2 Combine like terms: 4a2 + -4a2 = 0 1 + 2a + -4a2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -0.25 + -0.5a + a2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -0.5a + 0.25 + a2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -0.5a + a2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -0.5a + a2 = 0 + 0.25 -0.5a + a2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -0.5a + a2 = 0.25 The a term is -0.5a. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5a + 0.0625 + a2 = 0.25 + 0.0625 Reorder the terms: 0.0625 + -0.5a + a2 = 0.25 + 0.0625 Combine like terms: 0.25 + 0.0625 = 0.3125 0.0625 + -0.5a + a2 = 0.3125 Factor a perfect square on the left side: (a + -0.25)(a + -0.25) = 0.3125 Calculate the square root of the right side: 0.559016994 Break this problem into two subproblems by setting (a + -0.25) equal to 0.559016994 and -0.559016994.Subproblem 1
a + -0.25 = 0.559016994 Simplifying a + -0.25 = 0.559016994 Reorder the terms: -0.25 + a = 0.559016994 Solving -0.25 + a = 0.559016994 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + a = 0.559016994 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = 0.559016994 + 0.25 a = 0.559016994 + 0.25 Combine like terms: 0.559016994 + 0.25 = 0.809016994 a = 0.809016994 Simplifying a = 0.809016994Subproblem 2
a + -0.25 = -0.559016994 Simplifying a + -0.25 = -0.559016994 Reorder the terms: -0.25 + a = -0.559016994 Solving -0.25 + a = -0.559016994 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + a = -0.559016994 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = -0.559016994 + 0.25 a = -0.559016994 + 0.25 Combine like terms: -0.559016994 + 0.25 = -0.309016994 a = -0.309016994 Simplifying a = -0.309016994Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.809016994, -0.309016994}
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