4n^2-28n=-47

Solution for 4n^2-28n=-47 equation:

Simplifying
4n2 + -28n = -47

Reorder the terms:
-28n + 4n2 = -47

Solving
-28n + 4n2 = -47

Solving for variable 'n'.

Reorder the terms:
47 + -28n + 4n2 = -47 + 47

Combine like terms: -47 + 47 = 0
47 + -28n + 4n2 = 0

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

Divide each side by '4'.
11.75 + -7n + n2 = 0

Move the constant term to the right:

Add '-11.75' to each side of the equation.
11.75 + -7n + -11.75 + n2 = 0 + -11.75

Reorder the terms:
11.75 + -11.75 + -7n + n2 = 0 + -11.75

Combine like terms: 11.75 + -11.75 = 0.00
0.00 + -7n + n2 = 0 + -11.75
-7n + n2 = 0 + -11.75

Combine like terms: 0 + -11.75 = -11.75
-7n + n2 = -11.75

The n term is -7n.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7n + 12.25 + n2 = -11.75 + 12.25

Reorder the terms:
12.25 + -7n + n2 = -11.75 + 12.25

Combine like terms: -11.75 + 12.25 = 0.5
12.25 + -7n + n2 = 0.5

Factor a perfect square on the left side:
(n + -3.5)(n + -3.5) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(n + -3.5) equal to 0.707106781 and -0.707106781.

Subproblem 1
n + -3.5 = 0.707106781

Simplifying
n + -3.5 = 0.707106781

Reorder the terms:
-3.5 + n = 0.707106781

Solving
-3.5 + n = 0.707106781

Solving for variable 'n'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + n = 0.707106781 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + n = 0.707106781 + 3.5
n = 0.707106781 + 3.5

Combine like terms: 0.707106781 + 3.5 = 4.207106781
n = 4.207106781

Simplifying
n = 4.207106781

Subproblem 2
n + -3.5 = -0.707106781

Simplifying
n + -3.5 = -0.707106781

Reorder the terms:
-3.5 + n = -0.707106781

Solving
-3.5 + n = -0.707106781

Solving for variable 'n'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + n = -0.707106781 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + n = -0.707106781 + 3.5
n = -0.707106781 + 3.5

Combine like terms: -0.707106781 + 3.5 = 2.792893219
n = 2.792893219

Simplifying
n = 2.792893219

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {4.207106781, 2.792893219}