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Simplifying (4a2 + -3a + 8) + (6a2 + -18a + -15) = 0 Reorder the terms: (8 + -3a + 4a2) + (6a2 + -18a + -15) = 0 Remove parenthesis around (8 + -3a + 4a2) 8 + -3a + 4a2 + (6a2 + -18a + -15) = 0 Reorder the terms: 8 + -3a + 4a2 + (-15 + -18a + 6a2) = 0 Remove parenthesis around (-15 + -18a + 6a2) 8 + -3a + 4a2 + -15 + -18a + 6a2 = 0 Reorder the terms: 8 + -15 + -3a + -18a + 4a2 + 6a2 = 0 Combine like terms: 8 + -15 = -7 -7 + -3a + -18a + 4a2 + 6a2 = 0 Combine like terms: -3a + -18a = -21a -7 + -21a + 4a2 + 6a2 = 0 Combine like terms: 4a2 + 6a2 = 10a2 -7 + -21a + 10a2 = 0 Solving -7 + -21a + 10a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -0.7 + -2.1a + a2 = 0.0 Move the constant term to the right: Add '0.7' to each side of the equation. -0.7 + -2.1a + 0.7 + a2 = 0.0 + 0.7 Reorder the terms: -0.7 + 0.7 + -2.1a + a2 = 0.0 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + -2.1a + a2 = 0.0 + 0.7 -2.1a + a2 = 0.0 + 0.7 Combine like terms: 0.0 + 0.7 = 0.7 -2.1a + a2 = 0.7 The a term is -2.1a. Take half its coefficient (-1.05). Square it (1.1025) and add it to both sides. Add '1.1025' to each side of the equation. -2.1a + 1.1025 + a2 = 0.7 + 1.1025 Reorder the terms: 1.1025 + -2.1a + a2 = 0.7 + 1.1025 Combine like terms: 0.7 + 1.1025 = 1.8025 1.1025 + -2.1a + a2 = 1.8025 Factor a perfect square on the left side: (a + -1.05)(a + -1.05) = 1.8025 Calculate the square root of the right side: 1.342572158 Break this problem into two subproblems by setting (a + -1.05) equal to 1.342572158 and -1.342572158.Subproblem 1
a + -1.05 = 1.342572158 Simplifying a + -1.05 = 1.342572158 Reorder the terms: -1.05 + a = 1.342572158 Solving -1.05 + a = 1.342572158 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.05' to each side of the equation. -1.05 + 1.05 + a = 1.342572158 + 1.05 Combine like terms: -1.05 + 1.05 = 0.00 0.00 + a = 1.342572158 + 1.05 a = 1.342572158 + 1.05 Combine like terms: 1.342572158 + 1.05 = 2.392572158 a = 2.392572158 Simplifying a = 2.392572158Subproblem 2
a + -1.05 = -1.342572158 Simplifying a + -1.05 = -1.342572158 Reorder the terms: -1.05 + a = -1.342572158 Solving -1.05 + a = -1.342572158 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.05' to each side of the equation. -1.05 + 1.05 + a = -1.342572158 + 1.05 Combine like terms: -1.05 + 1.05 = 0.00 0.00 + a = -1.342572158 + 1.05 a = -1.342572158 + 1.05 Combine like terms: -1.342572158 + 1.05 = -0.292572158 a = -0.292572158 Simplifying a = -0.292572158Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.392572158, -0.292572158}
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