-4x^2-22x+31=0

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Solution for -4x^2-22x+31=0 equation: 


Simplifying
-4x2 + -22x + 31 = 0

Reorder the terms:
31 + -22x + -4x2 = 0

Solving
31 + -22x + -4x2 = 0

Solving for variable 'x'.

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

Divide each side by '-4'.
-7.75 + 5.5x + x2 = 0

Move the constant term to the right:

Add '7.75' to each side of the equation.
-7.75 + 5.5x + 7.75 + x2 = 0 + 7.75

Reorder the terms:
-7.75 + 7.75 + 5.5x + x2 = 0 + 7.75

Combine like terms: -7.75 + 7.75 = 0.00
0.00 + 5.5x + x2 = 0 + 7.75
5.5x + x2 = 0 + 7.75

Combine like terms: 0 + 7.75 = 7.75
5.5x + x2 = 7.75

The x term is 5.5x.  Take half its coefficient (2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
5.5x + 7.5625 + x2 = 7.75 + 7.5625

Reorder the terms:
7.5625 + 5.5x + x2 = 7.75 + 7.5625

Combine like terms: 7.75 + 7.5625 = 15.3125
7.5625 + 5.5x + x2 = 15.3125

Factor a perfect square on the left side:
(x + 2.75)(x + 2.75) = 15.3125

Calculate the square root of the right side: 3.913118961

Break this problem into two subproblems by setting 
(x + 2.75) equal to 3.913118961 and -3.913118961.


Subproblem 1
x + 2.75 = 3.913118961

Simplifying
x + 2.75 = 3.913118961

Reorder the terms:
2.75 + x = 3.913118961

Solving
2.75 + x = 3.913118961

Solving for variable 'x'.

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

Add '-2.75' to each side of the equation.
2.75 + -2.75 + x = 3.913118961 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + x = 3.913118961 + -2.75
x = 3.913118961 + -2.75

Combine like terms: 3.913118961 + -2.75 = 1.163118961
x = 1.163118961

Simplifying
x = 1.163118961


Subproblem 2
x + 2.75 = -3.913118961

Simplifying
x + 2.75 = -3.913118961

Reorder the terms:
2.75 + x = -3.913118961

Solving
2.75 + x = -3.913118961

Solving for variable 'x'.

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

Add '-2.75' to each side of the equation.
2.75 + -2.75 + x = -3.913118961 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + x = -3.913118961 + -2.75
x = -3.913118961 + -2.75

Combine like terms: -3.913118961 + -2.75 = -6.663118961
x = -6.663118961

Simplifying
x = -6.663118961


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.163118961, -6.663118961}

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