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Simplifying (25 + -1a)(2a + -3) = 40 Reorder the terms: (25 + -1a)(-3 + 2a) = 40 Multiply (25 + -1a) * (-3 + 2a) (25(-3 + 2a) + -1a * (-3 + 2a)) = 40 ((-3 * 25 + 2a * 25) + -1a * (-3 + 2a)) = 40 ((-75 + 50a) + -1a * (-3 + 2a)) = 40 (-75 + 50a + (-3 * -1a + 2a * -1a)) = 40 (-75 + 50a + (3a + -2a2)) = 40 Combine like terms: 50a + 3a = 53a (-75 + 53a + -2a2) = 40 Solving -75 + 53a + -2a2 = 40 Solving for variable 'a'. Reorder the terms: -75 + -40 + 53a + -2a2 = 40 + -40 Combine like terms: -75 + -40 = -115 -115 + 53a + -2a2 = 40 + -40 Combine like terms: 40 + -40 = 0 -115 + 53a + -2a2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 57.5 + -26.5a + a2 = 0 Move the constant term to the right: Add '-57.5' to each side of the equation. 57.5 + -26.5a + -57.5 + a2 = 0 + -57.5 Reorder the terms: 57.5 + -57.5 + -26.5a + a2 = 0 + -57.5 Combine like terms: 57.5 + -57.5 = 0.0 0.0 + -26.5a + a2 = 0 + -57.5 -26.5a + a2 = 0 + -57.5 Combine like terms: 0 + -57.5 = -57.5 -26.5a + a2 = -57.5 The a term is -26.5a. Take half its coefficient (-13.25). Square it (175.5625) and add it to both sides. Add '175.5625' to each side of the equation. -26.5a + 175.5625 + a2 = -57.5 + 175.5625 Reorder the terms: 175.5625 + -26.5a + a2 = -57.5 + 175.5625 Combine like terms: -57.5 + 175.5625 = 118.0625 175.5625 + -26.5a + a2 = 118.0625 Factor a perfect square on the left side: (a + -13.25)(a + -13.25) = 118.0625 Calculate the square root of the right side: 10.865656906 Break this problem into two subproblems by setting (a + -13.25) equal to 10.865656906 and -10.865656906.Subproblem 1
a + -13.25 = 10.865656906 Simplifying a + -13.25 = 10.865656906 Reorder the terms: -13.25 + a = 10.865656906 Solving -13.25 + a = 10.865656906 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '13.25' to each side of the equation. -13.25 + 13.25 + a = 10.865656906 + 13.25 Combine like terms: -13.25 + 13.25 = 0.00 0.00 + a = 10.865656906 + 13.25 a = 10.865656906 + 13.25 Combine like terms: 10.865656906 + 13.25 = 24.115656906 a = 24.115656906 Simplifying a = 24.115656906Subproblem 2
a + -13.25 = -10.865656906 Simplifying a + -13.25 = -10.865656906 Reorder the terms: -13.25 + a = -10.865656906 Solving -13.25 + a = -10.865656906 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '13.25' to each side of the equation. -13.25 + 13.25 + a = -10.865656906 + 13.25 Combine like terms: -13.25 + 13.25 = 0.00 0.00 + a = -10.865656906 + 13.25 a = -10.865656906 + 13.25 Combine like terms: -10.865656906 + 13.25 = 2.384343094 a = 2.384343094 Simplifying a = 2.384343094Solution
The solution to the problem is based on the solutions from the subproblems. a = {24.115656906, 2.384343094}
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