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Simplifying 18 = a(a + -1) Reorder the terms: 18 = a(-1 + a) 18 = (-1 * a + a * a) 18 = (-1a + a2) Solving 18 = -1a + a2 Solving for variable 'a'. Reorder the terms: 18 + a + -1a2 = -1a + a + a2 + -1a2 Combine like terms: -1a + a = 0 18 + a + -1a2 = 0 + a2 + -1a2 18 + a + -1a2 = a2 + -1a2 Combine like terms: a2 + -1a2 = 0 18 + a + -1a2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -18 + -1a + a2 = 0 Move the constant term to the right: Add '18' to each side of the equation. -18 + -1a + 18 + a2 = 0 + 18 Reorder the terms: -18 + 18 + -1a + a2 = 0 + 18 Combine like terms: -18 + 18 = 0 0 + -1a + a2 = 0 + 18 -1a + a2 = 0 + 18 Combine like terms: 0 + 18 = 18 -1a + a2 = 18 The a term is a. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1a + 0.25 + a2 = 18 + 0.25 Reorder the terms: 0.25 + -1a + a2 = 18 + 0.25 Combine like terms: 18 + 0.25 = 18.25 0.25 + -1a + a2 = 18.25 Factor a perfect square on the left side: (a + 0.5)(a + 0.5) = 18.25 Calculate the square root of the right side: 4.272001873 Break this problem into two subproblems by setting (a + 0.5) equal to 4.272001873 and -4.272001873.Subproblem 1
a + 0.5 = 4.272001873 Simplifying a + 0.5 = 4.272001873 Reorder the terms: 0.5 + a = 4.272001873 Solving 0.5 + a = 4.272001873 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 4.272001873 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 4.272001873 + -0.5 a = 4.272001873 + -0.5 Combine like terms: 4.272001873 + -0.5 = 3.772001873 a = 3.772001873 Simplifying a = 3.772001873Subproblem 2
a + 0.5 = -4.272001873 Simplifying a + 0.5 = -4.272001873 Reorder the terms: 0.5 + a = -4.272001873 Solving 0.5 + a = -4.272001873 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -4.272001873 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -4.272001873 + -0.5 a = -4.272001873 + -0.5 Combine like terms: -4.272001873 + -0.5 = -4.772001873 a = -4.772001873 Simplifying a = -4.772001873Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.772001873, -4.772001873}
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