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Simplifying (a + -5)(2a + 3) + -3(a + -4) = 50 Reorder the terms: (-5 + a)(2a + 3) + -3(a + -4) = 50 Reorder the terms: (-5 + a)(3 + 2a) + -3(a + -4) = 50 Multiply (-5 + a) * (3 + 2a) (-5(3 + 2a) + a(3 + 2a)) + -3(a + -4) = 50 ((3 * -5 + 2a * -5) + a(3 + 2a)) + -3(a + -4) = 50 ((-15 + -10a) + a(3 + 2a)) + -3(a + -4) = 50 (-15 + -10a + (3 * a + 2a * a)) + -3(a + -4) = 50 (-15 + -10a + (3a + 2a2)) + -3(a + -4) = 50 Combine like terms: -10a + 3a = -7a (-15 + -7a + 2a2) + -3(a + -4) = 50 Reorder the terms: -15 + -7a + 2a2 + -3(-4 + a) = 50 -15 + -7a + 2a2 + (-4 * -3 + a * -3) = 50 -15 + -7a + 2a2 + (12 + -3a) = 50 Reorder the terms: -15 + 12 + -7a + -3a + 2a2 = 50 Combine like terms: -15 + 12 = -3 -3 + -7a + -3a + 2a2 = 50 Combine like terms: -7a + -3a = -10a -3 + -10a + 2a2 = 50 Solving -3 + -10a + 2a2 = 50 Solving for variable 'a'. Reorder the terms: -3 + -50 + -10a + 2a2 = 50 + -50 Combine like terms: -3 + -50 = -53 -53 + -10a + 2a2 = 50 + -50 Combine like terms: 50 + -50 = 0 -53 + -10a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -26.5 + -5a + a2 = 0 Move the constant term to the right: Add '26.5' to each side of the equation. -26.5 + -5a + 26.5 + a2 = 0 + 26.5 Reorder the terms: -26.5 + 26.5 + -5a + a2 = 0 + 26.5 Combine like terms: -26.5 + 26.5 = 0.0 0.0 + -5a + a2 = 0 + 26.5 -5a + a2 = 0 + 26.5 Combine like terms: 0 + 26.5 = 26.5 -5a + a2 = 26.5 The a term is -5a. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5a + 6.25 + a2 = 26.5 + 6.25 Reorder the terms: 6.25 + -5a + a2 = 26.5 + 6.25 Combine like terms: 26.5 + 6.25 = 32.75 6.25 + -5a + a2 = 32.75 Factor a perfect square on the left side: (a + -2.5)(a + -2.5) = 32.75 Calculate the square root of the right side: 5.722761571 Break this problem into two subproblems by setting (a + -2.5) equal to 5.722761571 and -5.722761571.Subproblem 1
a + -2.5 = 5.722761571 Simplifying a + -2.5 = 5.722761571 Reorder the terms: -2.5 + a = 5.722761571 Solving -2.5 + a = 5.722761571 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + a = 5.722761571 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + a = 5.722761571 + 2.5 a = 5.722761571 + 2.5 Combine like terms: 5.722761571 + 2.5 = 8.222761571 a = 8.222761571 Simplifying a = 8.222761571Subproblem 2
a + -2.5 = -5.722761571 Simplifying a + -2.5 = -5.722761571 Reorder the terms: -2.5 + a = -5.722761571 Solving -2.5 + a = -5.722761571 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + a = -5.722761571 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + a = -5.722761571 + 2.5 a = -5.722761571 + 2.5 Combine like terms: -5.722761571 + 2.5 = -3.222761571 a = -3.222761571 Simplifying a = -3.222761571Solution
The solution to the problem is based on the solutions from the subproblems. a = {8.222761571, -3.222761571}
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