20a^2b^2+10a^3b^2+5ab^2=

Solution for 20a^2b^2+10a^3b^2+5ab^2= equation:

Simplifying
20a2b2 + 10a3b2 + 5ab2 = 0

Reorder the terms:
5ab2 + 20a2b2 + 10a3b2 = 0

Solving
5ab2 + 20a2b2 + 10a3b2 = 0

Solving for variable 'a'.

Factor out the Greatest Common Factor (GCF), '5ab2'.
5ab2(1 + 4a + 2a2) = 0

Ignore the factor 5.
Subproblem 1
Set the factor 'ab2' equal to zero and attempt to solve:

Simplifying
ab2 = 0

Solving
ab2 = 0

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

Simplifying
ab2 = 0

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Subproblem 2
Set the factor '(1 + 4a + 2a2)' equal to zero and attempt to solve:

Simplifying
1 + 4a + 2a2 = 0

Solving
1 + 4a + 2a2 = 0

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

Divide each side by '2'.
0.5 + 2a + a2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 2a + -0.5 + a2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 2a + a2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 2a + a2 = 0 + -0.5
2a + a2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
2a + a2 = -0.5

The a term is 2a.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2a + 1 + a2 = -0.5 + 1

Reorder the terms:
1 + 2a + a2 = -0.5 + 1

Combine like terms: -0.5 + 1 = 0.5
1 + 2a + a2 = 0.5

Factor a perfect square on the left side:
(a + 1)(a + 1) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(a + 1) equal to 0.707106781 and -0.707106781.

Subproblem 1
a + 1 = 0.707106781

Simplifying
a + 1 = 0.707106781

Reorder the terms:
1 + a = 0.707106781

Solving
1 + a = 0.707106781

Solving for variable 'a'.

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

Add '-1' to each side of the equation.
1 + -1 + a = 0.707106781 + -1

Combine like terms: 1 + -1 = 0
0 + a = 0.707106781 + -1
a = 0.707106781 + -1

Combine like terms: 0.707106781 + -1 = -0.292893219
a = -0.292893219

Simplifying
a = -0.292893219

Subproblem 2
a + 1 = -0.707106781

Simplifying
a + 1 = -0.707106781

Reorder the terms:
1 + a = -0.707106781

Solving
1 + a = -0.707106781

Solving for variable 'a'.

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

Add '-1' to each side of the equation.
1 + -1 + a = -0.707106781 + -1

Combine like terms: 1 + -1 = 0
0 + a = -0.707106781 + -1
a = -0.707106781 + -1

Combine like terms: -0.707106781 + -1 = -1.707106781
a = -1.707106781

Simplifying
a = -1.707106781

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {-0.292893219, -1.707106781}
Solution
a = {-0.292893219, -1.707106781}