4a^2+10a-22=0

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

Simplifying
4a2 + 10a + -22 = 0

Reorder the terms:
-22 + 10a + 4a2 = 0

Solving
-22 + 10a + 4a2 = 0

Solving for variable 'a'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-11 + 5a + 2a2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-11 + 5a + 2a2)' equal to zero and attempt to solve:

Simplifying
-11 + 5a + 2a2 = 0

Solving
-11 + 5a + 2a2 = 0

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

Divide each side by '2'.
-5.5 + 2.5a + a2 = 0

Move the constant term to the right:

Add '5.5' to each side of the equation.
-5.5 + 2.5a + 5.5 + a2 = 0 + 5.5

Reorder the terms:
-5.5 + 5.5 + 2.5a + a2 = 0 + 5.5

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + 2.5a + a2 = 0 + 5.5
2.5a + a2 = 0 + 5.5

Combine like terms: 0 + 5.5 = 5.5
2.5a + a2 = 5.5

The a term is 2.5a.  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.
2.5a + 1.5625 + a2 = 5.5 + 1.5625

Reorder the terms:
1.5625 + 2.5a + a2 = 5.5 + 1.5625

Combine like terms: 5.5 + 1.5625 = 7.0625
1.5625 + 2.5a + a2 = 7.0625

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

Calculate the square root of the right side: 2.657536453

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

Subproblem 1
a + 1.25 = 2.657536453

Simplifying
a + 1.25 = 2.657536453

Reorder the terms:
1.25 + a = 2.657536453

Solving
1.25 + a = 2.657536453

Solving for variable 'a'.

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

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

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

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

Simplifying
a = 1.407536453

Subproblem 2
a + 1.25 = -2.657536453

Simplifying
a + 1.25 = -2.657536453

Reorder the terms:
1.25 + a = -2.657536453

Solving
1.25 + a = -2.657536453

Solving for variable 'a'.

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

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

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

Combine like terms: -2.657536453 + -1.25 = -3.907536453
a = -3.907536453

Simplifying
a = -3.907536453

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