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Simplifying (a + 3)(2a + -3) = 9 Reorder the terms: (3 + a)(2a + -3) = 9 Reorder the terms: (3 + a)(-3 + 2a) = 9 Multiply (3 + a) * (-3 + 2a) (3(-3 + 2a) + a(-3 + 2a)) = 9 ((-3 * 3 + 2a * 3) + a(-3 + 2a)) = 9 ((-9 + 6a) + a(-3 + 2a)) = 9 (-9 + 6a + (-3 * a + 2a * a)) = 9 (-9 + 6a + (-3a + 2a2)) = 9 Combine like terms: 6a + -3a = 3a (-9 + 3a + 2a2) = 9 Solving -9 + 3a + 2a2 = 9 Solving for variable 'a'. Reorder the terms: -9 + -9 + 3a + 2a2 = 9 + -9 Combine like terms: -9 + -9 = -18 -18 + 3a + 2a2 = 9 + -9 Combine like terms: 9 + -9 = 0 -18 + 3a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -9 + 1.5a + a2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + 1.5a + 9 + a2 = 0 + 9 Reorder the terms: -9 + 9 + 1.5a + a2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + 1.5a + a2 = 0 + 9 1.5a + a2 = 0 + 9 Combine like terms: 0 + 9 = 9 1.5a + a2 = 9 The a term is 1.5a. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5a + 0.5625 + a2 = 9 + 0.5625 Reorder the terms: 0.5625 + 1.5a + a2 = 9 + 0.5625 Combine like terms: 9 + 0.5625 = 9.5625 0.5625 + 1.5a + a2 = 9.5625 Factor a perfect square on the left side: (a + 0.75)(a + 0.75) = 9.5625 Calculate the square root of the right side: 3.092329219 Break this problem into two subproblems by setting (a + 0.75) equal to 3.092329219 and -3.092329219.Subproblem 1
a + 0.75 = 3.092329219 Simplifying a + 0.75 = 3.092329219 Reorder the terms: 0.75 + a = 3.092329219 Solving 0.75 + a = 3.092329219 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = 3.092329219 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = 3.092329219 + -0.75 a = 3.092329219 + -0.75 Combine like terms: 3.092329219 + -0.75 = 2.342329219 a = 2.342329219 Simplifying a = 2.342329219Subproblem 2
a + 0.75 = -3.092329219 Simplifying a + 0.75 = -3.092329219 Reorder the terms: 0.75 + a = -3.092329219 Solving 0.75 + a = -3.092329219 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = -3.092329219 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = -3.092329219 + -0.75 a = -3.092329219 + -0.75 Combine like terms: -3.092329219 + -0.75 = -3.842329219 a = -3.842329219 Simplifying a = -3.842329219Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.342329219, -3.842329219}
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