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Simplifying 5a + -2a2 = 1 Solving 5a + -2a2 = 1 Solving for variable 'a'. Reorder the terms: -1 + 5a + -2a2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 5a + -2a2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 0.5 + -2.5a + a2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -2.5a + -0.5 + a2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -2.5a + a2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -2.5a + a2 = 0 + -0.5 -2.5a + a2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -2.5a + a2 = -0.5 The a term is -2.5a. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5a + 1.5625 + a2 = -0.5 + 1.5625 Reorder the terms: 1.5625 + -2.5a + a2 = -0.5 + 1.5625 Combine like terms: -0.5 + 1.5625 = 1.0625 1.5625 + -2.5a + a2 = 1.0625 Factor a perfect square on the left side: (a + -1.25)(a + -1.25) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (a + -1.25) equal to 1.030776406 and -1.030776406.Subproblem 1
a + -1.25 = 1.030776406 Simplifying a + -1.25 = 1.030776406 Reorder the terms: -1.25 + a = 1.030776406 Solving -1.25 + a = 1.030776406 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + a = 1.030776406 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + a = 1.030776406 + 1.25 a = 1.030776406 + 1.25 Combine like terms: 1.030776406 + 1.25 = 2.280776406 a = 2.280776406 Simplifying a = 2.280776406Subproblem 2
a + -1.25 = -1.030776406 Simplifying a + -1.25 = -1.030776406 Reorder the terms: -1.25 + a = -1.030776406 Solving -1.25 + a = -1.030776406 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + a = -1.030776406 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + a = -1.030776406 + 1.25 a = -1.030776406 + 1.25 Combine like terms: -1.030776406 + 1.25 = 0.219223594 a = 0.219223594 Simplifying a = 0.219223594Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.280776406, 0.219223594}
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