x(2+4x)=80

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Solution for x(2+4x)=80 equation: 


Simplifying
x(2 + 4x) = 80
(2 * x + 4x * x) = 80
(2x + 4x2) = 80

Solving
2x + 4x2 = 80

Solving for variable 'x'.

Reorder the terms:
-80 + 2x + 4x2 = 80 + -80

Combine like terms: 80 + -80 = 0
-80 + 2x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-40 + x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-40 + x + 2x2)' equal to zero and attempt to solve:

Simplifying
-40 + x + 2x2 = 0

Solving
-40 + x + 2x2 = 0

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

Divide each side by '2'.
-20 + 0.5x + x2 = 0

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + 0.5x + 20 + x2 = 0 + 20

Reorder the terms:
-20 + 20 + 0.5x + x2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + 0.5x + x2 = 0 + 20
0.5x + x2 = 0 + 20

Combine like terms: 0 + 20 = 20
0.5x + x2 = 20

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.5x + 0.25 + x2 = 20 + 0.25

Reorder the terms:
0.25 + 0.5x + x2 = 20 + 0.25

Combine like terms: 20 + 0.25 = 20.25
0.25 + 0.5x + x2 = 20.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 20.25

Calculate the square root of the right side: 4.5

Break this problem into two subproblems by setting 
(x + 0.5) equal to 4.5 and -4.5.


Subproblem 1
x + 0.5 = 4.5

Simplifying
x + 0.5 = 4.5

Reorder the terms:
0.5 + x = 4.5

Solving
0.5 + x = 4.5

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 4.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 4.5 + -0.5
x = 4.5 + -0.5

Combine like terms: 4.5 + -0.5 = 4
x = 4

Simplifying
x = 4


Subproblem 2
x + 0.5 = -4.5

Simplifying
x + 0.5 = -4.5

Reorder the terms:
0.5 + x = -4.5

Solving
0.5 + x = -4.5

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -4.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -4.5 + -0.5
x = -4.5 + -0.5

Combine like terms: -4.5 + -0.5 = -5
x = -5

Simplifying
x = -5


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4, -5}
Solution
x = {4, -5}

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