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Simplifying 140 + -20a + -4a2 = 0 Solving 140 + -20a + -4a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '4'. 4(35 + -5a + -1a2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(35 + -5a + -1a2)' equal to zero and attempt to solve: Simplifying 35 + -5a + -1a2 = 0 Solving 35 + -5a + -1a2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -35 + 5a + a2 = 0 Move the constant term to the right: Add '35' to each side of the equation. -35 + 5a + 35 + a2 = 0 + 35 Reorder the terms: -35 + 35 + 5a + a2 = 0 + 35 Combine like terms: -35 + 35 = 0 0 + 5a + a2 = 0 + 35 5a + a2 = 0 + 35 Combine like terms: 0 + 35 = 35 5a + a2 = 35 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 = 35 + 6.25 Reorder the terms: 6.25 + 5a + a2 = 35 + 6.25 Combine like terms: 35 + 6.25 = 41.25 6.25 + 5a + a2 = 41.25 Factor a perfect square on the left side: (a + 2.5)(a + 2.5) = 41.25 Calculate the square root of the right side: 6.422616289 Break this problem into two subproblems by setting (a + 2.5) equal to 6.422616289 and -6.422616289.Subproblem 1
a + 2.5 = 6.422616289 Simplifying a + 2.5 = 6.422616289 Reorder the terms: 2.5 + a = 6.422616289 Solving 2.5 + a = 6.422616289 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 = 6.422616289 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + a = 6.422616289 + -2.5 a = 6.422616289 + -2.5 Combine like terms: 6.422616289 + -2.5 = 3.922616289 a = 3.922616289 Simplifying a = 3.922616289Subproblem 2
a + 2.5 = -6.422616289 Simplifying a + 2.5 = -6.422616289 Reorder the terms: 2.5 + a = -6.422616289 Solving 2.5 + a = -6.422616289 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 = -6.422616289 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + a = -6.422616289 + -2.5 a = -6.422616289 + -2.5 Combine like terms: -6.422616289 + -2.5 = -8.922616289 a = -8.922616289 Simplifying a = -8.922616289Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.922616289, -8.922616289}Solution
a = {3.922616289, -8.922616289}
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