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Simplifying 2a(a + -1) = 3a + 40 Reorder the terms: 2a(-1 + a) = 3a + 40 (-1 * 2a + a * 2a) = 3a + 40 (-2a + 2a2) = 3a + 40 Reorder the terms: -2a + 2a2 = 40 + 3a Solving -2a + 2a2 = 40 + 3a Solving for variable 'a'. Reorder the terms: -40 + -2a + -3a + 2a2 = 40 + 3a + -40 + -3a Combine like terms: -2a + -3a = -5a -40 + -5a + 2a2 = 40 + 3a + -40 + -3a Reorder the terms: -40 + -5a + 2a2 = 40 + -40 + 3a + -3a Combine like terms: 40 + -40 = 0 -40 + -5a + 2a2 = 0 + 3a + -3a -40 + -5a + 2a2 = 3a + -3a Combine like terms: 3a + -3a = 0 -40 + -5a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -20 + -2.5a + a2 = 0 Move the constant term to the right: Add '20' to each side of the equation. -20 + -2.5a + 20 + a2 = 0 + 20 Reorder the terms: -20 + 20 + -2.5a + a2 = 0 + 20 Combine like terms: -20 + 20 = 0 0 + -2.5a + a2 = 0 + 20 -2.5a + a2 = 0 + 20 Combine like terms: 0 + 20 = 20 -2.5a + a2 = 20 The a term is -2.5a. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5a + 1.5625 + a2 = 20 + 1.5625 Reorder the terms: 1.5625 + -2.5a + a2 = 20 + 1.5625 Combine like terms: 20 + 1.5625 = 21.5625 1.5625 + -2.5a + a2 = 21.5625 Factor a perfect square on the left side: (a + -1.25)(a + -1.25) = 21.5625 Calculate the square root of the right side: 4.643543905 Break this problem into two subproblems by setting (a + -1.25) equal to 4.643543905 and -4.643543905.Subproblem 1
a + -1.25 = 4.643543905 Simplifying a + -1.25 = 4.643543905 Reorder the terms: -1.25 + a = 4.643543905 Solving -1.25 + a = 4.643543905 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + a = 4.643543905 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + a = 4.643543905 + 1.25 a = 4.643543905 + 1.25 Combine like terms: 4.643543905 + 1.25 = 5.893543905 a = 5.893543905 Simplifying a = 5.893543905Subproblem 2
a + -1.25 = -4.643543905 Simplifying a + -1.25 = -4.643543905 Reorder the terms: -1.25 + a = -4.643543905 Solving -1.25 + a = -4.643543905 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + a = -4.643543905 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + a = -4.643543905 + 1.25 a = -4.643543905 + 1.25 Combine like terms: -4.643543905 + 1.25 = -3.393543905 a = -3.393543905 Simplifying a = -3.393543905Solution
The solution to the problem is based on the solutions from the subproblems. a = {5.893543905, -3.393543905}
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