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Simplifying a2 + 8a + 2 = a + -5 Reorder the terms: 2 + 8a + a2 = a + -5 Reorder the terms: 2 + 8a + a2 = -5 + a Solving 2 + 8a + a2 = -5 + a Solving for variable 'a'. Reorder the terms: 2 + 5 + 8a + -1a + a2 = -5 + a + 5 + -1a Combine like terms: 2 + 5 = 7 7 + 8a + -1a + a2 = -5 + a + 5 + -1a Combine like terms: 8a + -1a = 7a 7 + 7a + a2 = -5 + a + 5 + -1a Reorder the terms: 7 + 7a + a2 = -5 + 5 + a + -1a Combine like terms: -5 + 5 = 0 7 + 7a + a2 = 0 + a + -1a 7 + 7a + a2 = a + -1a Combine like terms: a + -1a = 0 7 + 7a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '-7' to each side of the equation. 7 + 7a + -7 + a2 = 0 + -7 Reorder the terms: 7 + -7 + 7a + a2 = 0 + -7 Combine like terms: 7 + -7 = 0 0 + 7a + a2 = 0 + -7 7a + a2 = 0 + -7 Combine like terms: 0 + -7 = -7 7a + a2 = -7 The a term is 7a. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7a + 12.25 + a2 = -7 + 12.25 Reorder the terms: 12.25 + 7a + a2 = -7 + 12.25 Combine like terms: -7 + 12.25 = 5.25 12.25 + 7a + a2 = 5.25 Factor a perfect square on the left side: (a + 3.5)(a + 3.5) = 5.25 Calculate the square root of the right side: 2.291287847 Break this problem into two subproblems by setting (a + 3.5) equal to 2.291287847 and -2.291287847.Subproblem 1
a + 3.5 = 2.291287847 Simplifying a + 3.5 = 2.291287847 Reorder the terms: 3.5 + a = 2.291287847 Solving 3.5 + a = 2.291287847 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + a = 2.291287847 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + a = 2.291287847 + -3.5 a = 2.291287847 + -3.5 Combine like terms: 2.291287847 + -3.5 = -1.208712153 a = -1.208712153 Simplifying a = -1.208712153Subproblem 2
a + 3.5 = -2.291287847 Simplifying a + 3.5 = -2.291287847 Reorder the terms: 3.5 + a = -2.291287847 Solving 3.5 + a = -2.291287847 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + a = -2.291287847 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + a = -2.291287847 + -3.5 a = -2.291287847 + -3.5 Combine like terms: -2.291287847 + -3.5 = -5.791287847 a = -5.791287847 Simplifying a = -5.791287847Solution
The solution to the problem is based on the solutions from the subproblems. a = {-1.208712153, -5.791287847}
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