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Simplifying 2a2 + a + -40 = 0 Reorder the terms: -40 + a + 2a2 = 0 Solving -40 + a + 2a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -20 + 0.5a + a2 = 0 Move the constant term to the right: Add '20' to each side of the equation. -20 + 0.5a + 20 + a2 = 0 + 20 Reorder the terms: -20 + 20 + 0.5a + a2 = 0 + 20 Combine like terms: -20 + 20 = 0 0 + 0.5a + a2 = 0 + 20 0.5a + a2 = 0 + 20 Combine like terms: 0 + 20 = 20 0.5a + a2 = 20 The a term is a. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.5a + 0.25 + a2 = 20 + 0.25 Reorder the terms: 0.25 + 0.5a + a2 = 20 + 0.25 Combine like terms: 20 + 0.25 = 20.25 0.25 + 0.5a + a2 = 20.25 Factor a perfect square on the left side: (a + 0.5)(a + 0.5) = 20.25 Calculate the square root of the right side: 4.5 Break this problem into two subproblems by setting (a + 0.5) equal to 4.5 and -4.5.Subproblem 1
a + 0.5 = 4.5 Simplifying a + 0.5 = 4.5 Reorder the terms: 0.5 + a = 4.5 Solving 0.5 + a = 4.5 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 4.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 4.5 + -0.5 a = 4.5 + -0.5 Combine like terms: 4.5 + -0.5 = 4 a = 4 Simplifying a = 4Subproblem 2
a + 0.5 = -4.5 Simplifying a + 0.5 = -4.5 Reorder the terms: 0.5 + a = -4.5 Solving 0.5 + a = -4.5 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -4.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -4.5 + -0.5 a = -4.5 + -0.5 Combine like terms: -4.5 + -0.5 = -5 a = -5 Simplifying a = -5Solution
The solution to the problem is based on the solutions from the subproblems. a = {4, -5}
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