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Simplifying 0 = 5(2a2 + 475a + 3500) Reorder the terms: 0 = 5(3500 + 475a + 2a2) 0 = (3500 * 5 + 475a * 5 + 2a2 * 5) 0 = (17500 + 2375a + 10a2) Solving 0 = 17500 + 2375a + 10a2 Solving for variable 'a'. Combine like terms: 0 + -17500 = -17500 -17500 + -2375a + -10a2 = 17500 + 2375a + 10a2 + -17500 + -2375a + -10a2 Reorder the terms: -17500 + -2375a + -10a2 = 17500 + -17500 + 2375a + -2375a + 10a2 + -10a2 Combine like terms: 17500 + -17500 = 0 -17500 + -2375a + -10a2 = 0 + 2375a + -2375a + 10a2 + -10a2 -17500 + -2375a + -10a2 = 2375a + -2375a + 10a2 + -10a2 Combine like terms: 2375a + -2375a = 0 -17500 + -2375a + -10a2 = 0 + 10a2 + -10a2 -17500 + -2375a + -10a2 = 10a2 + -10a2 Combine like terms: 10a2 + -10a2 = 0 -17500 + -2375a + -10a2 = 0 Factor out the Greatest Common Factor (GCF), '-5'. -5(3500 + 475a + 2a2) = 0 Ignore the factor -5.Subproblem 1
Set the factor '(3500 + 475a + 2a2)' equal to zero and attempt to solve: Simplifying 3500 + 475a + 2a2 = 0 Solving 3500 + 475a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1750 + 237.5a + a2 = 0 Move the constant term to the right: Add '-1750' to each side of the equation. 1750 + 237.5a + -1750 + a2 = 0 + -1750 Reorder the terms: 1750 + -1750 + 237.5a + a2 = 0 + -1750 Combine like terms: 1750 + -1750 = 0 0 + 237.5a + a2 = 0 + -1750 237.5a + a2 = 0 + -1750 Combine like terms: 0 + -1750 = -1750 237.5a + a2 = -1750 The a term is 237.5a. Take half its coefficient (118.75). Square it (14101.5625) and add it to both sides. Add '14101.5625' to each side of the equation. 237.5a + 14101.5625 + a2 = -1750 + 14101.5625 Reorder the terms: 14101.5625 + 237.5a + a2 = -1750 + 14101.5625 Combine like terms: -1750 + 14101.5625 = 12351.5625 14101.5625 + 237.5a + a2 = 12351.5625 Factor a perfect square on the left side: (a + 118.75)(a + 118.75) = 12351.5625 Calculate the square root of the right side: 111.137583652 Break this problem into two subproblems by setting (a + 118.75) equal to 111.137583652 and -111.137583652.Subproblem 1
a + 118.75 = 111.137583652 Simplifying a + 118.75 = 111.137583652 Reorder the terms: 118.75 + a = 111.137583652 Solving 118.75 + a = 111.137583652 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-118.75' to each side of the equation. 118.75 + -118.75 + a = 111.137583652 + -118.75 Combine like terms: 118.75 + -118.75 = 0.00 0.00 + a = 111.137583652 + -118.75 a = 111.137583652 + -118.75 Combine like terms: 111.137583652 + -118.75 = -7.612416348 a = -7.612416348 Simplifying a = -7.612416348Subproblem 2
a + 118.75 = -111.137583652 Simplifying a + 118.75 = -111.137583652 Reorder the terms: 118.75 + a = -111.137583652 Solving 118.75 + a = -111.137583652 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-118.75' to each side of the equation. 118.75 + -118.75 + a = -111.137583652 + -118.75 Combine like terms: 118.75 + -118.75 = 0.00 0.00 + a = -111.137583652 + -118.75 a = -111.137583652 + -118.75 Combine like terms: -111.137583652 + -118.75 = -229.887583652 a = -229.887583652 Simplifying a = -229.887583652Solution
The solution to the problem is based on the solutions from the subproblems. a = {-7.612416348, -229.887583652}Solution
a = {-7.612416348, -229.887583652}
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