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Simplifying 2a * a + 4a + -16 = a + -1 Multiply a * a 2a2 + 4a + -16 = a + -1 Reorder the terms: -16 + 4a + 2a2 = a + -1 Reorder the terms: -16 + 4a + 2a2 = -1 + a Solving -16 + 4a + 2a2 = -1 + a Solving for variable 'a'. Reorder the terms: -16 + 1 + 4a + -1a + 2a2 = -1 + a + 1 + -1a Combine like terms: -16 + 1 = -15 -15 + 4a + -1a + 2a2 = -1 + a + 1 + -1a Combine like terms: 4a + -1a = 3a -15 + 3a + 2a2 = -1 + a + 1 + -1a Reorder the terms: -15 + 3a + 2a2 = -1 + 1 + a + -1a Combine like terms: -1 + 1 = 0 -15 + 3a + 2a2 = 0 + a + -1a -15 + 3a + 2a2 = a + -1a Combine like terms: a + -1a = 0 -15 + 3a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -7.5 + 1.5a + a2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + 1.5a + 7.5 + a2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + 1.5a + a2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + 1.5a + a2 = 0 + 7.5 1.5a + a2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 1.5a + a2 = 7.5 The a term is 1.5a. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5a + 0.5625 + a2 = 7.5 + 0.5625 Reorder the terms: 0.5625 + 1.5a + a2 = 7.5 + 0.5625 Combine like terms: 7.5 + 0.5625 = 8.0625 0.5625 + 1.5a + a2 = 8.0625 Factor a perfect square on the left side: (a + 0.75)(a + 0.75) = 8.0625 Calculate the square root of the right side: 2.839454173 Break this problem into two subproblems by setting (a + 0.75) equal to 2.839454173 and -2.839454173.Subproblem 1
a + 0.75 = 2.839454173 Simplifying a + 0.75 = 2.839454173 Reorder the terms: 0.75 + a = 2.839454173 Solving 0.75 + a = 2.839454173 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = 2.839454173 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = 2.839454173 + -0.75 a = 2.839454173 + -0.75 Combine like terms: 2.839454173 + -0.75 = 2.089454173 a = 2.089454173 Simplifying a = 2.089454173Subproblem 2
a + 0.75 = -2.839454173 Simplifying a + 0.75 = -2.839454173 Reorder the terms: 0.75 + a = -2.839454173 Solving 0.75 + a = -2.839454173 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = -2.839454173 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = -2.839454173 + -0.75 a = -2.839454173 + -0.75 Combine like terms: -2.839454173 + -0.75 = -3.589454173 a = -3.589454173 Simplifying a = -3.589454173Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.089454173, -3.589454173}
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