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Simplifying 2a2 + -15a = 180 Reorder the terms: -15a + 2a2 = 180 Solving -15a + 2a2 = 180 Solving for variable 'a'. Reorder the terms: -180 + -15a + 2a2 = 180 + -180 Combine like terms: 180 + -180 = 0 -180 + -15a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -90 + -7.5a + a2 = 0 Move the constant term to the right: Add '90' to each side of the equation. -90 + -7.5a + 90 + a2 = 0 + 90 Reorder the terms: -90 + 90 + -7.5a + a2 = 0 + 90 Combine like terms: -90 + 90 = 0 0 + -7.5a + a2 = 0 + 90 -7.5a + a2 = 0 + 90 Combine like terms: 0 + 90 = 90 -7.5a + a2 = 90 The a term is -7.5a. Take half its coefficient (-3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. -7.5a + 14.0625 + a2 = 90 + 14.0625 Reorder the terms: 14.0625 + -7.5a + a2 = 90 + 14.0625 Combine like terms: 90 + 14.0625 = 104.0625 14.0625 + -7.5a + a2 = 104.0625 Factor a perfect square on the left side: (a + -3.75)(a + -3.75) = 104.0625 Calculate the square root of the right side: 10.201102882 Break this problem into two subproblems by setting (a + -3.75) equal to 10.201102882 and -10.201102882.Subproblem 1
a + -3.75 = 10.201102882 Simplifying a + -3.75 = 10.201102882 Reorder the terms: -3.75 + a = 10.201102882 Solving -3.75 + a = 10.201102882 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + a = 10.201102882 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + a = 10.201102882 + 3.75 a = 10.201102882 + 3.75 Combine like terms: 10.201102882 + 3.75 = 13.951102882 a = 13.951102882 Simplifying a = 13.951102882Subproblem 2
a + -3.75 = -10.201102882 Simplifying a + -3.75 = -10.201102882 Reorder the terms: -3.75 + a = -10.201102882 Solving -3.75 + a = -10.201102882 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + a = -10.201102882 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + a = -10.201102882 + 3.75 a = -10.201102882 + 3.75 Combine like terms: -10.201102882 + 3.75 = -6.451102882 a = -6.451102882 Simplifying a = -6.451102882Solution
The solution to the problem is based on the solutions from the subproblems. a = {13.951102882, -6.451102882}
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