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Simplifying 2a + 2(a + -80) * a + (a + -80) = 240 Reorder the terms: 2a + 2(-80 + a) * a + (a + -80) = 240 Reorder the terms for easier multiplication: 2a + 2a(-80 + a) + (a + -80) = 240 2a + (-80 * 2a + a * 2a) + (a + -80) = 240 2a + (-160a + 2a2) + (a + -80) = 240 Reorder the terms: 2a + -160a + 2a2 + (-80 + a) = 240 Remove parenthesis around (-80 + a) 2a + -160a + 2a2 + -80 + a = 240 Reorder the terms: -80 + 2a + -160a + a + 2a2 = 240 Combine like terms: 2a + -160a = -158a -80 + -158a + a + 2a2 = 240 Combine like terms: -158a + a = -157a -80 + -157a + 2a2 = 240 Solving -80 + -157a + 2a2 = 240 Solving for variable 'a'. Reorder the terms: -80 + -240 + -157a + 2a2 = 240 + -240 Combine like terms: -80 + -240 = -320 -320 + -157a + 2a2 = 240 + -240 Combine like terms: 240 + -240 = 0 -320 + -157a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -160 + -78.5a + a2 = 0 Move the constant term to the right: Add '160' to each side of the equation. -160 + -78.5a + 160 + a2 = 0 + 160 Reorder the terms: -160 + 160 + -78.5a + a2 = 0 + 160 Combine like terms: -160 + 160 = 0 0 + -78.5a + a2 = 0 + 160 -78.5a + a2 = 0 + 160 Combine like terms: 0 + 160 = 160 -78.5a + a2 = 160 The a term is -78.5a. Take half its coefficient (-39.25). Square it (1540.5625) and add it to both sides. Add '1540.5625' to each side of the equation. -78.5a + 1540.5625 + a2 = 160 + 1540.5625 Reorder the terms: 1540.5625 + -78.5a + a2 = 160 + 1540.5625 Combine like terms: 160 + 1540.5625 = 1700.5625 1540.5625 + -78.5a + a2 = 1700.5625 Factor a perfect square on the left side: (a + -39.25)(a + -39.25) = 1700.5625 Calculate the square root of the right side: 41.237877006 Break this problem into two subproblems by setting (a + -39.25) equal to 41.237877006 and -41.237877006.Subproblem 1
a + -39.25 = 41.237877006 Simplifying a + -39.25 = 41.237877006 Reorder the terms: -39.25 + a = 41.237877006 Solving -39.25 + a = 41.237877006 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '39.25' to each side of the equation. -39.25 + 39.25 + a = 41.237877006 + 39.25 Combine like terms: -39.25 + 39.25 = 0.00 0.00 + a = 41.237877006 + 39.25 a = 41.237877006 + 39.25 Combine like terms: 41.237877006 + 39.25 = 80.487877006 a = 80.487877006 Simplifying a = 80.487877006Subproblem 2
a + -39.25 = -41.237877006 Simplifying a + -39.25 = -41.237877006 Reorder the terms: -39.25 + a = -41.237877006 Solving -39.25 + a = -41.237877006 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '39.25' to each side of the equation. -39.25 + 39.25 + a = -41.237877006 + 39.25 Combine like terms: -39.25 + 39.25 = 0.00 0.00 + a = -41.237877006 + 39.25 a = -41.237877006 + 39.25 Combine like terms: -41.237877006 + 39.25 = -1.987877006 a = -1.987877006 Simplifying a = -1.987877006Solution
The solution to the problem is based on the solutions from the subproblems. a = {80.487877006, -1.987877006}
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