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Simplifying (0.4 + -9a)(0.1 + -5a) = 0 Multiply (0.4 + -9a) * (0.1 + -5a) (0.4(0.1 + -5a) + -9a * (0.1 + -5a)) = 0 ((0.1 * 0.4 + -5a * 0.4) + -9a * (0.1 + -5a)) = 0 ((0.04 + -2a) + -9a * (0.1 + -5a)) = 0 (0.04 + -2a + (0.1 * -9a + -5a * -9a)) = 0 (0.04 + -2a + (-0.9a + 45a2)) = 0 Combine like terms: -2a + -0.9a = -2.9a (0.04 + -2.9a + 45a2) = 0 Solving 0.04 + -2.9a + 45a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 45 the coefficient of the squared term: Divide each side by '45'. 0.0008888888889 + -0.06444444444a + a2 = 0 Move the constant term to the right: Add '-0.0008888888889' to each side of the equation. 0.0008888888889 + -0.06444444444a + -0.0008888888889 + a2 = 0 + -0.0008888888889 Reorder the terms: 0.0008888888889 + -0.0008888888889 + -0.06444444444a + a2 = 0 + -0.0008888888889 Combine like terms: 0.0008888888889 + -0.0008888888889 = 0.0000000000000 0.0000000000000 + -0.06444444444a + a2 = 0 + -0.0008888888889 -0.06444444444a + a2 = 0 + -0.0008888888889 Combine like terms: 0 + -0.0008888888889 = -0.0008888888889 -0.06444444444a + a2 = -0.0008888888889 The a term is -0.06444444444a. Take half its coefficient (-0.03222222222). Square it (0.001038271605) and add it to both sides. Add '0.001038271605' to each side of the equation. -0.06444444444a + 0.001038271605 + a2 = -0.0008888888889 + 0.001038271605 Reorder the terms: 0.001038271605 + -0.06444444444a + a2 = -0.0008888888889 + 0.001038271605 Combine like terms: -0.0008888888889 + 0.001038271605 = 0.0001493827161 0.001038271605 + -0.06444444444a + a2 = 0.0001493827161 Factor a perfect square on the left side: (a + -0.03222222222)(a + -0.03222222222) = 0.0001493827161 Calculate the square root of the right side: 0.012222222 Break this problem into two subproblems by setting (a + -0.03222222222) equal to 0.012222222 and -0.012222222.Subproblem 1
a + -0.03222222222 = 0.012222222 Simplifying a + -0.03222222222 = 0.012222222 Reorder the terms: -0.03222222222 + a = 0.012222222 Solving -0.03222222222 + a = 0.012222222 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.03222222222' to each side of the equation. -0.03222222222 + 0.03222222222 + a = 0.012222222 + 0.03222222222 Combine like terms: -0.03222222222 + 0.03222222222 = 0.00000000000 0.00000000000 + a = 0.012222222 + 0.03222222222 a = 0.012222222 + 0.03222222222 Combine like terms: 0.012222222 + 0.03222222222 = 0.04444444422 a = 0.04444444422 Simplifying a = 0.04444444422Subproblem 2
a + -0.03222222222 = -0.012222222 Simplifying a + -0.03222222222 = -0.012222222 Reorder the terms: -0.03222222222 + a = -0.012222222 Solving -0.03222222222 + a = -0.012222222 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.03222222222' to each side of the equation. -0.03222222222 + 0.03222222222 + a = -0.012222222 + 0.03222222222 Combine like terms: -0.03222222222 + 0.03222222222 = 0.00000000000 0.00000000000 + a = -0.012222222 + 0.03222222222 a = -0.012222222 + 0.03222222222 Combine like terms: -0.012222222 + 0.03222222222 = 0.02000000022 a = 0.02000000022 Simplifying a = 0.02000000022Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.04444444422, 0.02000000022}
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