4x^2-10x-22=0

Solution for 4x^2-10x-22=0 equation:

Simplifying
4x2 + -10x + -22 = 0

Reorder the terms:
-22 + -10x + 4x2 = 0

Solving
-22 + -10x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-11 + -5x + 2x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-11 + -5x + 2x2)' equal to zero and attempt to solve:

Simplifying
-11 + -5x + 2x2 = 0

Solving
-11 + -5x + 2x2 = 0

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

Divide each side by '2'.
-5.5 + -2.5x + x2 = 0

Move the constant term to the right:

Add '5.5' to each side of the equation.
-5.5 + -2.5x + 5.5 + x2 = 0 + 5.5

Reorder the terms:
-5.5 + 5.5 + -2.5x + x2 = 0 + 5.5

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + -2.5x + x2 = 0 + 5.5
-2.5x + x2 = 0 + 5.5

Combine like terms: 0 + 5.5 = 5.5
-2.5x + x2 = 5.5

The x term is -2.5x.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5x + 1.5625 + x2 = 5.5 + 1.5625

Reorder the terms:
1.5625 + -2.5x + x2 = 5.5 + 1.5625

Combine like terms: 5.5 + 1.5625 = 7.0625
1.5625 + -2.5x + x2 = 7.0625

Factor a perfect square on the left side:
(x + -1.25)(x + -1.25) = 7.0625

Calculate the square root of the right side: 2.657536453

Break this problem into two subproblems by setting 
(x + -1.25) equal to 2.657536453 and -2.657536453.

Subproblem 1
x + -1.25 = 2.657536453

Simplifying
x + -1.25 = 2.657536453

Reorder the terms:
-1.25 + x = 2.657536453

Solving
-1.25 + x = 2.657536453

Solving for variable 'x'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + x = 2.657536453 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x = 2.657536453 + 1.25
x = 2.657536453 + 1.25

Combine like terms: 2.657536453 + 1.25 = 3.907536453
x = 3.907536453

Simplifying
x = 3.907536453

Subproblem 2
x + -1.25 = -2.657536453

Simplifying
x + -1.25 = -2.657536453

Reorder the terms:
-1.25 + x = -2.657536453

Solving
-1.25 + x = -2.657536453

Solving for variable 'x'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + x = -2.657536453 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x = -2.657536453 + 1.25
x = -2.657536453 + 1.25

Combine like terms: -2.657536453 + 1.25 = -1.407536453
x = -1.407536453

Simplifying
x = -1.407536453

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.907536453, -1.407536453}
Solution
x = {3.907536453, -1.407536453}