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Simplifying (0.5a + -0.1)(0.09a + 1) = 0 Reorder the terms: (-0.1 + 0.5a)(0.09a + 1) = 0 Reorder the terms: (-0.1 + 0.5a)(1 + 0.09a) = 0 Multiply (-0.1 + 0.5a) * (1 + 0.09a) (-0.1(1 + 0.09a) + 0.5a * (1 + 0.09a)) = 0 ((1 * -0.1 + 0.09a * -0.1) + 0.5a * (1 + 0.09a)) = 0 ((-0.1 + -0.009a) + 0.5a * (1 + 0.09a)) = 0 (-0.1 + -0.009a + (1 * 0.5a + 0.09a * 0.5a)) = 0 (-0.1 + -0.009a + (0.5a + 0.045a2)) = 0 Combine like terms: -0.009a + 0.5a = 0.491a (-0.1 + 0.491a + 0.045a2) = 0 Solving -0.1 + 0.491a + 0.045a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 0.045 the coefficient of the squared term: Divide each side by '0.045'. -2.222222222 + 10.91111111a + a2 = 0 Move the constant term to the right: Add '2.222222222' to each side of the equation. -2.222222222 + 10.91111111a + 2.222222222 + a2 = 0 + 2.222222222 Reorder the terms: -2.222222222 + 2.222222222 + 10.91111111a + a2 = 0 + 2.222222222 Combine like terms: -2.222222222 + 2.222222222 = 0.000000000 0.000000000 + 10.91111111a + a2 = 0 + 2.222222222 10.91111111a + a2 = 0 + 2.222222222 Combine like terms: 0 + 2.222222222 = 2.222222222 10.91111111a + a2 = 2.222222222 The a term is 10.91111111a. Take half its coefficient (5.455555555). Square it (29.76308641) and add it to both sides. Add '29.76308641' to each side of the equation. 10.91111111a + 29.76308641 + a2 = 2.222222222 + 29.76308641 Reorder the terms: 29.76308641 + 10.91111111a + a2 = 2.222222222 + 29.76308641 Combine like terms: 2.222222222 + 29.76308641 = 31.985308632 29.76308641 + 10.91111111a + a2 = 31.985308632 Factor a perfect square on the left side: (a + 5.455555555)(a + 5.455555555) = 31.985308632 Calculate the square root of the right side: 5.655555555 Break this problem into two subproblems by setting (a + 5.455555555) equal to 5.655555555 and -5.655555555.Subproblem 1
a + 5.455555555 = 5.655555555 Simplifying a + 5.455555555 = 5.655555555 Reorder the terms: 5.455555555 + a = 5.655555555 Solving 5.455555555 + a = 5.655555555 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-5.455555555' to each side of the equation. 5.455555555 + -5.455555555 + a = 5.655555555 + -5.455555555 Combine like terms: 5.455555555 + -5.455555555 = 0.000000000 0.000000000 + a = 5.655555555 + -5.455555555 a = 5.655555555 + -5.455555555 Combine like terms: 5.655555555 + -5.455555555 = 0.2 a = 0.2 Simplifying a = 0.2Subproblem 2
a + 5.455555555 = -5.655555555 Simplifying a + 5.455555555 = -5.655555555 Reorder the terms: 5.455555555 + a = -5.655555555 Solving 5.455555555 + a = -5.655555555 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-5.455555555' to each side of the equation. 5.455555555 + -5.455555555 + a = -5.655555555 + -5.455555555 Combine like terms: 5.455555555 + -5.455555555 = 0.000000000 0.000000000 + a = -5.655555555 + -5.455555555 a = -5.655555555 + -5.455555555 Combine like terms: -5.655555555 + -5.455555555 = -11.11111111 a = -11.11111111 Simplifying a = -11.11111111Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.2, -11.11111111}
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