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Simplifying 82a2 + -38a = 279 Reorder the terms: -38a + 82a2 = 279 Solving -38a + 82a2 = 279 Solving for variable 'a'. Reorder the terms: -279 + -38a + 82a2 = 279 + -279 Combine like terms: 279 + -279 = 0 -279 + -38a + 82a2 = 0 Begin completing the square. Divide all terms by 82 the coefficient of the squared term: Divide each side by '82'. -3.402439024 + -0.4634146341a + a2 = 0 Move the constant term to the right: Add '3.402439024' to each side of the equation. -3.402439024 + -0.4634146341a + 3.402439024 + a2 = 0 + 3.402439024 Reorder the terms: -3.402439024 + 3.402439024 + -0.4634146341a + a2 = 0 + 3.402439024 Combine like terms: -3.402439024 + 3.402439024 = 0.000000000 0.000000000 + -0.4634146341a + a2 = 0 + 3.402439024 -0.4634146341a + a2 = 0 + 3.402439024 Combine like terms: 0 + 3.402439024 = 3.402439024 -0.4634146341a + a2 = 3.402439024 The a term is -0.4634146341a. Take half its coefficient (-0.2317073171). Square it (0.05368828080) and add it to both sides. Add '0.05368828080' to each side of the equation. -0.4634146341a + 0.05368828080 + a2 = 3.402439024 + 0.05368828080 Reorder the terms: 0.05368828080 + -0.4634146341a + a2 = 3.402439024 + 0.05368828080 Combine like terms: 3.402439024 + 0.05368828080 = 3.4561273048 0.05368828080 + -0.4634146341a + a2 = 3.4561273048 Factor a perfect square on the left side: (a + -0.2317073171)(a + -0.2317073171) = 3.4561273048 Calculate the square root of the right side: 1.859066245 Break this problem into two subproblems by setting (a + -0.2317073171) equal to 1.859066245 and -1.859066245.Subproblem 1
a + -0.2317073171 = 1.859066245 Simplifying a + -0.2317073171 = 1.859066245 Reorder the terms: -0.2317073171 + a = 1.859066245 Solving -0.2317073171 + a = 1.859066245 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.2317073171' to each side of the equation. -0.2317073171 + 0.2317073171 + a = 1.859066245 + 0.2317073171 Combine like terms: -0.2317073171 + 0.2317073171 = 0.0000000000 0.0000000000 + a = 1.859066245 + 0.2317073171 a = 1.859066245 + 0.2317073171 Combine like terms: 1.859066245 + 0.2317073171 = 2.0907735621 a = 2.0907735621 Simplifying a = 2.0907735621Subproblem 2
a + -0.2317073171 = -1.859066245 Simplifying a + -0.2317073171 = -1.859066245 Reorder the terms: -0.2317073171 + a = -1.859066245 Solving -0.2317073171 + a = -1.859066245 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.2317073171' to each side of the equation. -0.2317073171 + 0.2317073171 + a = -1.859066245 + 0.2317073171 Combine like terms: -0.2317073171 + 0.2317073171 = 0.0000000000 0.0000000000 + a = -1.859066245 + 0.2317073171 a = -1.859066245 + 0.2317073171 Combine like terms: -1.859066245 + 0.2317073171 = -1.6273589279 a = -1.6273589279 Simplifying a = -1.6273589279Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.0907735621, -1.6273589279}
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