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Simplifying 2a2 + -6a + -18 = 1 Reorder the terms: -18 + -6a + 2a2 = 1 Solving -18 + -6a + 2a2 = 1 Solving for variable 'a'. Reorder the terms: -18 + -1 + -6a + 2a2 = 1 + -1 Combine like terms: -18 + -1 = -19 -19 + -6a + 2a2 = 1 + -1 Combine like terms: 1 + -1 = 0 -19 + -6a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -9.5 + -3a + a2 = 0 Move the constant term to the right: Add '9.5' to each side of the equation. -9.5 + -3a + 9.5 + a2 = 0 + 9.5 Reorder the terms: -9.5 + 9.5 + -3a + a2 = 0 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + -3a + a2 = 0 + 9.5 -3a + a2 = 0 + 9.5 Combine like terms: 0 + 9.5 = 9.5 -3a + a2 = 9.5 The a term is -3a. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3a + 2.25 + a2 = 9.5 + 2.25 Reorder the terms: 2.25 + -3a + a2 = 9.5 + 2.25 Combine like terms: 9.5 + 2.25 = 11.75 2.25 + -3a + a2 = 11.75 Factor a perfect square on the left side: (a + -1.5)(a + -1.5) = 11.75 Calculate the square root of the right side: 3.4278273 Break this problem into two subproblems by setting (a + -1.5) equal to 3.4278273 and -3.4278273.Subproblem 1
a + -1.5 = 3.4278273 Simplifying a + -1.5 = 3.4278273 Reorder the terms: -1.5 + a = 3.4278273 Solving -1.5 + a = 3.4278273 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + a = 3.4278273 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + a = 3.4278273 + 1.5 a = 3.4278273 + 1.5 Combine like terms: 3.4278273 + 1.5 = 4.9278273 a = 4.9278273 Simplifying a = 4.9278273Subproblem 2
a + -1.5 = -3.4278273 Simplifying a + -1.5 = -3.4278273 Reorder the terms: -1.5 + a = -3.4278273 Solving -1.5 + a = -3.4278273 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + a = -3.4278273 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + a = -3.4278273 + 1.5 a = -3.4278273 + 1.5 Combine like terms: -3.4278273 + 1.5 = -1.9278273 a = -1.9278273 Simplifying a = -1.9278273Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.9278273, -1.9278273}
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