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Simplifying -2(2a2 + 4a + -5) + 3(3a2 + -5a + 1) = 0 Reorder the terms: -2(-5 + 4a + 2a2) + 3(3a2 + -5a + 1) = 0 (-5 * -2 + 4a * -2 + 2a2 * -2) + 3(3a2 + -5a + 1) = 0 (10 + -8a + -4a2) + 3(3a2 + -5a + 1) = 0 Reorder the terms: 10 + -8a + -4a2 + 3(1 + -5a + 3a2) = 0 10 + -8a + -4a2 + (1 * 3 + -5a * 3 + 3a2 * 3) = 0 10 + -8a + -4a2 + (3 + -15a + 9a2) = 0 Reorder the terms: 10 + 3 + -8a + -15a + -4a2 + 9a2 = 0 Combine like terms: 10 + 3 = 13 13 + -8a + -15a + -4a2 + 9a2 = 0 Combine like terms: -8a + -15a = -23a 13 + -23a + -4a2 + 9a2 = 0 Combine like terms: -4a2 + 9a2 = 5a2 13 + -23a + 5a2 = 0 Solving 13 + -23a + 5a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 2.6 + -4.6a + a2 = 0 Move the constant term to the right: Add '-2.6' to each side of the equation. 2.6 + -4.6a + -2.6 + a2 = 0 + -2.6 Reorder the terms: 2.6 + -2.6 + -4.6a + a2 = 0 + -2.6 Combine like terms: 2.6 + -2.6 = 0.0 0.0 + -4.6a + a2 = 0 + -2.6 -4.6a + a2 = 0 + -2.6 Combine like terms: 0 + -2.6 = -2.6 -4.6a + a2 = -2.6 The a term is -4.6a. Take half its coefficient (-2.3). Square it (5.29) and add it to both sides. Add '5.29' to each side of the equation. -4.6a + 5.29 + a2 = -2.6 + 5.29 Reorder the terms: 5.29 + -4.6a + a2 = -2.6 + 5.29 Combine like terms: -2.6 + 5.29 = 2.69 5.29 + -4.6a + a2 = 2.69 Factor a perfect square on the left side: (a + -2.3)(a + -2.3) = 2.69 Calculate the square root of the right side: 1.640121947 Break this problem into two subproblems by setting (a + -2.3) equal to 1.640121947 and -1.640121947.Subproblem 1
a + -2.3 = 1.640121947 Simplifying a + -2.3 = 1.640121947 Reorder the terms: -2.3 + a = 1.640121947 Solving -2.3 + a = 1.640121947 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.3' to each side of the equation. -2.3 + 2.3 + a = 1.640121947 + 2.3 Combine like terms: -2.3 + 2.3 = 0.0 0.0 + a = 1.640121947 + 2.3 a = 1.640121947 + 2.3 Combine like terms: 1.640121947 + 2.3 = 3.940121947 a = 3.940121947 Simplifying a = 3.940121947Subproblem 2
a + -2.3 = -1.640121947 Simplifying a + -2.3 = -1.640121947 Reorder the terms: -2.3 + a = -1.640121947 Solving -2.3 + a = -1.640121947 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.3' to each side of the equation. -2.3 + 2.3 + a = -1.640121947 + 2.3 Combine like terms: -2.3 + 2.3 = 0.0 0.0 + a = -1.640121947 + 2.3 a = -1.640121947 + 2.3 Combine like terms: -1.640121947 + 2.3 = 0.659878053 a = 0.659878053 Simplifying a = 0.659878053Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.940121947, 0.659878053}
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