90(x)=50+10x+2x^2

Solution for 90(x)=50+10x+2x^2 equation:

Simplifying
90(x) = 50 + 10x + 2x2

Solving
90x = 50 + 10x + 2x2

Solving for variable 'x'.

Reorder the terms:
-50 + 90x + -10x + -2x2 = 50 + 10x + 2x2 + -50 + -10x + -2x2

Combine like terms: 90x + -10x = 80x
-50 + 80x + -2x2 = 50 + 10x + 2x2 + -50 + -10x + -2x2

Reorder the terms:
-50 + 80x + -2x2 = 50 + -50 + 10x + -10x + 2x2 + -2x2

Combine like terms: 50 + -50 = 0
-50 + 80x + -2x2 = 0 + 10x + -10x + 2x2 + -2x2
-50 + 80x + -2x2 = 10x + -10x + 2x2 + -2x2

Combine like terms: 10x + -10x = 0
-50 + 80x + -2x2 = 0 + 2x2 + -2x2
-50 + 80x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
-50 + 80x + -2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-25 + 40x + -1x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-25 + 40x + -1x2)' equal to zero and attempt to solve:

Simplifying
-25 + 40x + -1x2 = 0

Solving
-25 + 40x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
25 + -40x + x2 = 0

Move the constant term to the right:

Add '-25' to each side of the equation.
25 + -40x + -25 + x2 = 0 + -25

Reorder the terms:
25 + -25 + -40x + x2 = 0 + -25

Combine like terms: 25 + -25 = 0
0 + -40x + x2 = 0 + -25
-40x + x2 = 0 + -25

Combine like terms: 0 + -25 = -25
-40x + x2 = -25

The x term is -40x.  Take half its coefficient (-20).
Square it (400) and add it to both sides.

Add '400' to each side of the equation.
-40x + 400 + x2 = -25 + 400

Reorder the terms:
400 + -40x + x2 = -25 + 400

Combine like terms: -25 + 400 = 375
400 + -40x + x2 = 375

Factor a perfect square on the left side:
(x + -20)(x + -20) = 375

Calculate the square root of the right side: 19.364916731

Break this problem into two subproblems by setting 
(x + -20) equal to 19.364916731 and -19.364916731.

Subproblem 1
x + -20 = 19.364916731

Simplifying
x + -20 = 19.364916731

Reorder the terms:
-20 + x = 19.364916731

Solving
-20 + x = 19.364916731

Solving for variable 'x'.

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

Add '20' to each side of the equation.
-20 + 20 + x = 19.364916731 + 20

Combine like terms: -20 + 20 = 0
0 + x = 19.364916731 + 20
x = 19.364916731 + 20

Combine like terms: 19.364916731 + 20 = 39.364916731
x = 39.364916731

Simplifying
x = 39.364916731

Subproblem 2
x + -20 = -19.364916731

Simplifying
x + -20 = -19.364916731

Reorder the terms:
-20 + x = -19.364916731

Solving
-20 + x = -19.364916731

Solving for variable 'x'.

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

Add '20' to each side of the equation.
-20 + 20 + x = -19.364916731 + 20

Combine like terms: -20 + 20 = 0
0 + x = -19.364916731 + 20
x = -19.364916731 + 20

Combine like terms: -19.364916731 + 20 = 0.635083269
x = 0.635083269

Simplifying
x = 0.635083269

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {39.364916731, 0.635083269}
Solution
x = {39.364916731, 0.635083269}