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Simplifying 0 = 5(2a2 + -95a + 490) Reorder the terms: 0 = 5(490 + -95a + 2a2) 0 = (490 * 5 + -95a * 5 + 2a2 * 5) 0 = (2450 + -475a + 10a2) Solving 0 = 2450 + -475a + 10a2 Solving for variable 'a'. Combine like terms: 0 + -2450 = -2450 -2450 + 475a + -10a2 = 2450 + -475a + 10a2 + -2450 + 475a + -10a2 Reorder the terms: -2450 + 475a + -10a2 = 2450 + -2450 + -475a + 475a + 10a2 + -10a2 Combine like terms: 2450 + -2450 = 0 -2450 + 475a + -10a2 = 0 + -475a + 475a + 10a2 + -10a2 -2450 + 475a + -10a2 = -475a + 475a + 10a2 + -10a2 Combine like terms: -475a + 475a = 0 -2450 + 475a + -10a2 = 0 + 10a2 + -10a2 -2450 + 475a + -10a2 = 10a2 + -10a2 Combine like terms: 10a2 + -10a2 = 0 -2450 + 475a + -10a2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-490 + 95a + -2a2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-490 + 95a + -2a2)' equal to zero and attempt to solve: Simplifying -490 + 95a + -2a2 = 0 Solving -490 + 95a + -2a2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 245 + -47.5a + a2 = 0 Move the constant term to the right: Add '-245' to each side of the equation. 245 + -47.5a + -245 + a2 = 0 + -245 Reorder the terms: 245 + -245 + -47.5a + a2 = 0 + -245 Combine like terms: 245 + -245 = 0 0 + -47.5a + a2 = 0 + -245 -47.5a + a2 = 0 + -245 Combine like terms: 0 + -245 = -245 -47.5a + a2 = -245 The a term is -47.5a. Take half its coefficient (-23.75). Square it (564.0625) and add it to both sides. Add '564.0625' to each side of the equation. -47.5a + 564.0625 + a2 = -245 + 564.0625 Reorder the terms: 564.0625 + -47.5a + a2 = -245 + 564.0625 Combine like terms: -245 + 564.0625 = 319.0625 564.0625 + -47.5a + a2 = 319.0625 Factor a perfect square on the left side: (a + -23.75)(a + -23.75) = 319.0625 Calculate the square root of the right side: 17.862320678 Break this problem into two subproblems by setting (a + -23.75) equal to 17.862320678 and -17.862320678.Subproblem 1
a + -23.75 = 17.862320678 Simplifying a + -23.75 = 17.862320678 Reorder the terms: -23.75 + a = 17.862320678 Solving -23.75 + a = 17.862320678 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '23.75' to each side of the equation. -23.75 + 23.75 + a = 17.862320678 + 23.75 Combine like terms: -23.75 + 23.75 = 0.00 0.00 + a = 17.862320678 + 23.75 a = 17.862320678 + 23.75 Combine like terms: 17.862320678 + 23.75 = 41.612320678 a = 41.612320678 Simplifying a = 41.612320678Subproblem 2
a + -23.75 = -17.862320678 Simplifying a + -23.75 = -17.862320678 Reorder the terms: -23.75 + a = -17.862320678 Solving -23.75 + a = -17.862320678 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '23.75' to each side of the equation. -23.75 + 23.75 + a = -17.862320678 + 23.75 Combine like terms: -23.75 + 23.75 = 0.00 0.00 + a = -17.862320678 + 23.75 a = -17.862320678 + 23.75 Combine like terms: -17.862320678 + 23.75 = 5.887679322 a = 5.887679322 Simplifying a = 5.887679322Solution
The solution to the problem is based on the solutions from the subproblems. a = {41.612320678, 5.887679322}Solution
a = {41.612320678, 5.887679322}
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