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Simplifying 13a + a2 = 32 Solving 13a + a2 = 32 Solving for variable 'a'. Reorder the terms: -32 + 13a + a2 = 32 + -32 Combine like terms: 32 + -32 = 0 -32 + 13a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '32' to each side of the equation. -32 + 13a + 32 + a2 = 0 + 32 Reorder the terms: -32 + 32 + 13a + a2 = 0 + 32 Combine like terms: -32 + 32 = 0 0 + 13a + a2 = 0 + 32 13a + a2 = 0 + 32 Combine like terms: 0 + 32 = 32 13a + a2 = 32 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 = 32 + 42.25 Reorder the terms: 42.25 + 13a + a2 = 32 + 42.25 Combine like terms: 32 + 42.25 = 74.25 42.25 + 13a + a2 = 74.25 Factor a perfect square on the left side: (a + 6.5)(a + 6.5) = 74.25 Calculate the square root of the right side: 8.61684397 Break this problem into two subproblems by setting (a + 6.5) equal to 8.61684397 and -8.61684397.Subproblem 1
a + 6.5 = 8.61684397 Simplifying a + 6.5 = 8.61684397 Reorder the terms: 6.5 + a = 8.61684397 Solving 6.5 + a = 8.61684397 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 = 8.61684397 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + a = 8.61684397 + -6.5 a = 8.61684397 + -6.5 Combine like terms: 8.61684397 + -6.5 = 2.11684397 a = 2.11684397 Simplifying a = 2.11684397Subproblem 2
a + 6.5 = -8.61684397 Simplifying a + 6.5 = -8.61684397 Reorder the terms: 6.5 + a = -8.61684397 Solving 6.5 + a = -8.61684397 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 = -8.61684397 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + a = -8.61684397 + -6.5 a = -8.61684397 + -6.5 Combine like terms: -8.61684397 + -6.5 = -15.11684397 a = -15.11684397 Simplifying a = -15.11684397Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.11684397, -15.11684397}
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