4v^2-27v-34=0

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Solution for 4v^2-27v-34=0 equation: 


Simplifying
4v2 + -27v + -34 = 0

Reorder the terms:
-34 + -27v + 4v2 = 0

Solving
-34 + -27v + 4v2 = 0

Solving for variable 'v'.

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
-8.5 + -6.75v + v2 = 0

Move the constant term to the right:

Add '8.5' to each side of the equation.
-8.5 + -6.75v + 8.5 + v2 = 0 + 8.5

Reorder the terms:
-8.5 + 8.5 + -6.75v + v2 = 0 + 8.5

Combine like terms: -8.5 + 8.5 = 0.0
0.0 + -6.75v + v2 = 0 + 8.5
-6.75v + v2 = 0 + 8.5

Combine like terms: 0 + 8.5 = 8.5
-6.75v + v2 = 8.5

The v term is -6.75v.  Take half its coefficient (-3.375).
Square it (11.390625) and add it to both sides.

Add '11.390625' to each side of the equation.
-6.75v + 11.390625 + v2 = 8.5 + 11.390625

Reorder the terms:
11.390625 + -6.75v + v2 = 8.5 + 11.390625

Combine like terms: 8.5 + 11.390625 = 19.890625
11.390625 + -6.75v + v2 = 19.890625

Factor a perfect square on the left side:
(v + -3.375)(v + -3.375) = 19.890625

Calculate the square root of the right side: 4.459890694

Break this problem into two subproblems by setting 
(v + -3.375) equal to 4.459890694 and -4.459890694.


Subproblem 1
v + -3.375 = 4.459890694

Simplifying
v + -3.375 = 4.459890694

Reorder the terms:
-3.375 + v = 4.459890694

Solving
-3.375 + v = 4.459890694

Solving for variable 'v'.

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

Add '3.375' to each side of the equation.
-3.375 + 3.375 + v = 4.459890694 + 3.375

Combine like terms: -3.375 + 3.375 = 0.000
0.000 + v = 4.459890694 + 3.375
v = 4.459890694 + 3.375

Combine like terms: 4.459890694 + 3.375 = 7.834890694
v = 7.834890694

Simplifying
v = 7.834890694


Subproblem 2
v + -3.375 = -4.459890694

Simplifying
v + -3.375 = -4.459890694

Reorder the terms:
-3.375 + v = -4.459890694

Solving
-3.375 + v = -4.459890694

Solving for variable 'v'.

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

Add '3.375' to each side of the equation.
-3.375 + 3.375 + v = -4.459890694 + 3.375

Combine like terms: -3.375 + 3.375 = 0.000
0.000 + v = -4.459890694 + 3.375
v = -4.459890694 + 3.375

Combine like terms: -4.459890694 + 3.375 = -1.084890694
v = -1.084890694

Simplifying
v = -1.084890694


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {7.834890694, -1.084890694}

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