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Simplifying 2a[a + 3] = 3[a + 2] Reorder the terms: 2a[3 + a] = 3[a + 2] [3 * 2a + a * 2a] = 3[a + 2] [6a + 2a2] = 3[a + 2] Reorder the terms: 6a + 2a2 = 3[2 + a] 6a + 2a2 = [2 * 3 + a * 3] 6a + 2a2 = [6 + 3a] Solving 6a + 2a2 = 6 + 3a Solving for variable 'a'. Reorder the terms: -6 + 6a + -3a + 2a2 = 6 + 3a + -6 + -3a Combine like terms: 6a + -3a = 3a -6 + 3a + 2a2 = 6 + 3a + -6 + -3a Reorder the terms: -6 + 3a + 2a2 = 6 + -6 + 3a + -3a Combine like terms: 6 + -6 = 0 -6 + 3a + 2a2 = 0 + 3a + -3a -6 + 3a + 2a2 = 3a + -3a Combine like terms: 3a + -3a = 0 -6 + 3a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3 + 1.5a + a2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + 1.5a + 3 + a2 = 0 + 3 Reorder the terms: -3 + 3 + 1.5a + a2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 1.5a + a2 = 0 + 3 1.5a + a2 = 0 + 3 Combine like terms: 0 + 3 = 3 1.5a + a2 = 3 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 = 3 + 0.5625 Reorder the terms: 0.5625 + 1.5a + a2 = 3 + 0.5625 Combine like terms: 3 + 0.5625 = 3.5625 0.5625 + 1.5a + a2 = 3.5625 Factor a perfect square on the left side: (a + 0.75)(a + 0.75) = 3.5625 Calculate the square root of the right side: 1.887458609 Break this problem into two subproblems by setting (a + 0.75) equal to 1.887458609 and -1.887458609.Subproblem 1
a + 0.75 = 1.887458609 Simplifying a + 0.75 = 1.887458609 Reorder the terms: 0.75 + a = 1.887458609 Solving 0.75 + a = 1.887458609 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 = 1.887458609 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = 1.887458609 + -0.75 a = 1.887458609 + -0.75 Combine like terms: 1.887458609 + -0.75 = 1.137458609 a = 1.137458609 Simplifying a = 1.137458609Subproblem 2
a + 0.75 = -1.887458609 Simplifying a + 0.75 = -1.887458609 Reorder the terms: 0.75 + a = -1.887458609 Solving 0.75 + a = -1.887458609 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 = -1.887458609 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = -1.887458609 + -0.75 a = -1.887458609 + -0.75 Combine like terms: -1.887458609 + -0.75 = -2.637458609 a = -2.637458609 Simplifying a = -2.637458609Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.137458609, -2.637458609}
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