90=40+10Q+2Q^2

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Solution for 90=40+10Q+2Q^2 equation: 


Simplifying
90 = 40 + 10Q + 2Q2

Solving
90 = 40 + 10Q + 2Q2

Solving for variable 'Q'.

Combine like terms: 90 + -40 = 50
50 + -10Q + -2Q2 = 40 + 10Q + 2Q2 + -40 + -10Q + -2Q2

Reorder the terms:
50 + -10Q + -2Q2 = 40 + -40 + 10Q + -10Q + 2Q2 + -2Q2

Combine like terms: 40 + -40 = 0
50 + -10Q + -2Q2 = 0 + 10Q + -10Q + 2Q2 + -2Q2
50 + -10Q + -2Q2 = 10Q + -10Q + 2Q2 + -2Q2

Combine like terms: 10Q + -10Q = 0
50 + -10Q + -2Q2 = 0 + 2Q2 + -2Q2
50 + -10Q + -2Q2 = 2Q2 + -2Q2

Combine like terms: 2Q2 + -2Q2 = 0
50 + -10Q + -2Q2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(25 + -5Q + -1Q2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(25 + -5Q + -1Q2)' equal to zero and attempt to solve:

Simplifying
25 + -5Q + -1Q2 = 0

Solving
25 + -5Q + -1Q2 = 0

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

Divide each side by '-1'.
-25 + 5Q + Q2 = 0

Move the constant term to the right:

Add '25' to each side of the equation.
-25 + 5Q + 25 + Q2 = 0 + 25

Reorder the terms:
-25 + 25 + 5Q + Q2 = 0 + 25

Combine like terms: -25 + 25 = 0
0 + 5Q + Q2 = 0 + 25
5Q + Q2 = 0 + 25

Combine like terms: 0 + 25 = 25
5Q + Q2 = 25

The Q term is 5Q.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5Q + 6.25 + Q2 = 25 + 6.25

Reorder the terms:
6.25 + 5Q + Q2 = 25 + 6.25

Combine like terms: 25 + 6.25 = 31.25
6.25 + 5Q + Q2 = 31.25

Factor a perfect square on the left side:
(Q + 2.5)(Q + 2.5) = 31.25

Calculate the square root of the right side: 5.590169944

Break this problem into two subproblems by setting 
(Q + 2.5) equal to 5.590169944 and -5.590169944.


Subproblem 1
Q + 2.5 = 5.590169944

Simplifying
Q + 2.5 = 5.590169944

Reorder the terms:
2.5 + Q = 5.590169944

Solving
2.5 + Q = 5.590169944

Solving for variable 'Q'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + Q = 5.590169944 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + Q = 5.590169944 + -2.5
Q = 5.590169944 + -2.5

Combine like terms: 5.590169944 + -2.5 = 3.090169944
Q = 3.090169944

Simplifying
Q = 3.090169944


Subproblem 2
Q + 2.5 = -5.590169944

Simplifying
Q + 2.5 = -5.590169944

Reorder the terms:
2.5 + Q = -5.590169944

Solving
2.5 + Q = -5.590169944

Solving for variable 'Q'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + Q = -5.590169944 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + Q = -5.590169944 + -2.5
Q = -5.590169944 + -2.5

Combine like terms: -5.590169944 + -2.5 = -8.090169944
Q = -8.090169944

Simplifying
Q = -8.090169944


Solution
The solution to the problem is based on the solutions
from the subproblems.
Q = {3.090169944, -8.090169944}
Solution
Q = {3.090169944, -8.090169944}

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