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Simplifying 125 = 0.01a2 + 0.05a + 107 Reorder the terms: 125 = 107 + 0.05a + 0.01a2 Solving 125 = 107 + 0.05a + 0.01a2 Solving for variable 'a'. Combine like terms: 125 + -107 = 18 18 + -0.05a + -0.01a2 = 107 + 0.05a + 0.01a2 + -107 + -0.05a + -0.01a2 Reorder the terms: 18 + -0.05a + -0.01a2 = 107 + -107 + 0.05a + -0.05a + 0.01a2 + -0.01a2 Combine like terms: 107 + -107 = 0 18 + -0.05a + -0.01a2 = 0 + 0.05a + -0.05a + 0.01a2 + -0.01a2 18 + -0.05a + -0.01a2 = 0.05a + -0.05a + 0.01a2 + -0.01a2 Combine like terms: 0.05a + -0.05a = 0.00 18 + -0.05a + -0.01a2 = 0.00 + 0.01a2 + -0.01a2 18 + -0.05a + -0.01a2 = 0.01a2 + -0.01a2 Combine like terms: 0.01a2 + -0.01a2 = 0.00 18 + -0.05a + -0.01a2 = 0.00 Begin completing the square. Divide all terms by -0.01 the coefficient of the squared term: Divide each side by '-0.01'. -1800 + 5a + a2 = 0 Move the constant term to the right: Add '1800' to each side of the equation. -1800 + 5a + 1800 + a2 = 0 + 1800 Reorder the terms: -1800 + 1800 + 5a + a2 = 0 + 1800 Combine like terms: -1800 + 1800 = 0 0 + 5a + a2 = 0 + 1800 5a + a2 = 0 + 1800 Combine like terms: 0 + 1800 = 1800 5a + a2 = 1800 The a term is 5a. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5a + 6.25 + a2 = 1800 + 6.25 Reorder the terms: 6.25 + 5a + a2 = 1800 + 6.25 Combine like terms: 1800 + 6.25 = 1806.25 6.25 + 5a + a2 = 1806.25 Factor a perfect square on the left side: (a + 2.5)(a + 2.5) = 1806.25 Calculate the square root of the right side: 42.5 Break this problem into two subproblems by setting (a + 2.5) equal to 42.5 and -42.5.Subproblem 1
a + 2.5 = 42.5 Simplifying a + 2.5 = 42.5 Reorder the terms: 2.5 + a = 42.5 Solving 2.5 + a = 42.5 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + a = 42.5 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + a = 42.5 + -2.5 a = 42.5 + -2.5 Combine like terms: 42.5 + -2.5 = 40 a = 40 Simplifying a = 40Subproblem 2
a + 2.5 = -42.5 Simplifying a + 2.5 = -42.5 Reorder the terms: 2.5 + a = -42.5 Solving 2.5 + a = -42.5 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + a = -42.5 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + a = -42.5 + -2.5 a = -42.5 + -2.5 Combine like terms: -42.5 + -2.5 = -45 a = -45 Simplifying a = -45Solution
The solution to the problem is based on the solutions from the subproblems. a = {40, -45}
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