13a+a^2=32

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Solution for 13a+a^2=32 equation:


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 = 2.11684397

Subproblem 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 = -15.11684397

Solution

The solution to the problem is based on the solutions from the subproblems. a = {2.11684397, -15.11684397}

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