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Simplifying 9(1 + -1a2) + 3(a + -1) + -2 = 0 (1 * 9 + -1a2 * 9) + 3(a + -1) + -2 = 0 (9 + -9a2) + 3(a + -1) + -2 = 0 Reorder the terms: 9 + -9a2 + 3(-1 + a) + -2 = 0 9 + -9a2 + (-1 * 3 + a * 3) + -2 = 0 9 + -9a2 + (-3 + 3a) + -2 = 0 Reorder the terms: 9 + -3 + -2 + 3a + -9a2 = 0 Combine like terms: 9 + -3 = 6 6 + -2 + 3a + -9a2 = 0 Combine like terms: 6 + -2 = 4 4 + 3a + -9a2 = 0 Solving 4 + 3a + -9a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -0.4444444444 + -0.3333333333a + a2 = 0 Move the constant term to the right: Add '0.4444444444' to each side of the equation. -0.4444444444 + -0.3333333333a + 0.4444444444 + a2 = 0 + 0.4444444444 Reorder the terms: -0.4444444444 + 0.4444444444 + -0.3333333333a + a2 = 0 + 0.4444444444 Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000 0.0000000000 + -0.3333333333a + a2 = 0 + 0.4444444444 -0.3333333333a + a2 = 0 + 0.4444444444 Combine like terms: 0 + 0.4444444444 = 0.4444444444 -0.3333333333a + a2 = 0.4444444444 The a term is -0.3333333333a. Take half its coefficient (-0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. -0.3333333333a + 0.02777777779 + a2 = 0.4444444444 + 0.02777777779 Reorder the terms: 0.02777777779 + -0.3333333333a + a2 = 0.4444444444 + 0.02777777779 Combine like terms: 0.4444444444 + 0.02777777779 = 0.47222222219 0.02777777779 + -0.3333333333a + a2 = 0.47222222219 Factor a perfect square on the left side: (a + -0.1666666667)(a + -0.1666666667) = 0.47222222219 Calculate the square root of the right side: 0.687184271 Break this problem into two subproblems by setting (a + -0.1666666667) equal to 0.687184271 and -0.687184271.Subproblem 1
a + -0.1666666667 = 0.687184271 Simplifying a + -0.1666666667 = 0.687184271 Reorder the terms: -0.1666666667 + a = 0.687184271 Solving -0.1666666667 + a = 0.687184271 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + a = 0.687184271 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + a = 0.687184271 + 0.1666666667 a = 0.687184271 + 0.1666666667 Combine like terms: 0.687184271 + 0.1666666667 = 0.8538509377 a = 0.8538509377 Simplifying a = 0.8538509377Subproblem 2
a + -0.1666666667 = -0.687184271 Simplifying a + -0.1666666667 = -0.687184271 Reorder the terms: -0.1666666667 + a = -0.687184271 Solving -0.1666666667 + a = -0.687184271 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + a = -0.687184271 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + a = -0.687184271 + 0.1666666667 a = -0.687184271 + 0.1666666667 Combine like terms: -0.687184271 + 0.1666666667 = -0.5205176043 a = -0.5205176043 Simplifying a = -0.5205176043Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.8538509377, -0.5205176043}
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