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Simplifying a2 + 13a + -68 = 52 Reorder the terms: -68 + 13a + a2 = 52 Solving -68 + 13a + a2 = 52 Solving for variable 'a'. Reorder the terms: -68 + -52 + 13a + a2 = 52 + -52 Combine like terms: -68 + -52 = -120 -120 + 13a + a2 = 52 + -52 Combine like terms: 52 + -52 = 0 -120 + 13a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '120' to each side of the equation. -120 + 13a + 120 + a2 = 0 + 120 Reorder the terms: -120 + 120 + 13a + a2 = 0 + 120 Combine like terms: -120 + 120 = 0 0 + 13a + a2 = 0 + 120 13a + a2 = 0 + 120 Combine like terms: 0 + 120 = 120 13a + a2 = 120 The a term is 13a. Take half its coefficient (6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. 13a + 42.25 + a2 = 120 + 42.25 Reorder the terms: 42.25 + 13a + a2 = 120 + 42.25 Combine like terms: 120 + 42.25 = 162.25 42.25 + 13a + a2 = 162.25 Factor a perfect square on the left side: (a + 6.5)(a + 6.5) = 162.25 Calculate the square root of the right side: 12.737739203 Break this problem into two subproblems by setting (a + 6.5) equal to 12.737739203 and -12.737739203.Subproblem 1
a + 6.5 = 12.737739203 Simplifying a + 6.5 = 12.737739203 Reorder the terms: 6.5 + a = 12.737739203 Solving 6.5 + a = 12.737739203 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + a = 12.737739203 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + a = 12.737739203 + -6.5 a = 12.737739203 + -6.5 Combine like terms: 12.737739203 + -6.5 = 6.237739203 a = 6.237739203 Simplifying a = 6.237739203Subproblem 2
a + 6.5 = -12.737739203 Simplifying a + 6.5 = -12.737739203 Reorder the terms: 6.5 + a = -12.737739203 Solving 6.5 + a = -12.737739203 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + a = -12.737739203 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + a = -12.737739203 + -6.5 a = -12.737739203 + -6.5 Combine like terms: -12.737739203 + -6.5 = -19.237739203 a = -19.237739203 Simplifying a = -19.237739203Solution
The solution to the problem is based on the solutions from the subproblems. a = {6.237739203, -19.237739203}
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