v^2-14v-31=0

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Solution for v^2-14v-31=0 equation: 


Simplifying
v2 + -14v + -31 = 0

Reorder the terms:
-31 + -14v + v2 = 0

Solving
-31 + -14v + v2 = 0

Solving for variable 'v'.

Begin completing the square.

Move the constant term to the right:

Add '31' to each side of the equation.
-31 + -14v + 31 + v2 = 0 + 31

Reorder the terms:
-31 + 31 + -14v + v2 = 0 + 31

Combine like terms: -31 + 31 = 0
0 + -14v + v2 = 0 + 31
-14v + v2 = 0 + 31

Combine like terms: 0 + 31 = 31
-14v + v2 = 31

The v term is -14v.  Take half its coefficient (-7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
-14v + 49 + v2 = 31 + 49

Reorder the terms:
49 + -14v + v2 = 31 + 49

Combine like terms: 31 + 49 = 80
49 + -14v + v2 = 80

Factor a perfect square on the left side:
(v + -7)(v + -7) = 80

Calculate the square root of the right side: 8.94427191

Break this problem into two subproblems by setting 
(v + -7) equal to 8.94427191 and -8.94427191.


Subproblem 1
v + -7 = 8.94427191

Simplifying
v + -7 = 8.94427191

Reorder the terms:
-7 + v = 8.94427191

Solving
-7 + v = 8.94427191

Solving for variable 'v'.

Move all terms containing v to the left, all other terms to the right.

Add '7' to each side of the equation.
-7 + 7 + v = 8.94427191 + 7

Combine like terms: -7 + 7 = 0
0 + v = 8.94427191 + 7
v = 8.94427191 + 7

Combine like terms: 8.94427191 + 7 = 15.94427191
v = 15.94427191

Simplifying
v = 15.94427191


Subproblem 2
v + -7 = -8.94427191

Simplifying
v + -7 = -8.94427191

Reorder the terms:
-7 + v = -8.94427191

Solving
-7 + v = -8.94427191

Solving for variable 'v'.

Move all terms containing v to the left, all other terms to the right.

Add '7' to each side of the equation.
-7 + 7 + v = -8.94427191 + 7

Combine like terms: -7 + 7 = 0
0 + v = -8.94427191 + 7
v = -8.94427191 + 7

Combine like terms: -8.94427191 + 7 = -1.94427191
v = -1.94427191

Simplifying
v = -1.94427191


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {15.94427191, -1.94427191}

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