3q^2-24q+40=0

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Solution for 3q^2-24q+40=0 equation: 


Simplifying
3q2 + -24q + 40 = 0

Reorder the terms:
40 + -24q + 3q2 = 0

Solving
40 + -24q + 3q2 = 0

Solving for variable 'q'.

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

Divide each side by '3'.
13.33333333 + -8q + q2 = 0

Move the constant term to the right:

Add '-13.33333333' to each side of the equation.
13.33333333 + -8q + -13.33333333 + q2 = 0 + -13.33333333

Reorder the terms:
13.33333333 + -13.33333333 + -8q + q2 = 0 + -13.33333333

Combine like terms: 13.33333333 + -13.33333333 = 0.00000000
0.00000000 + -8q + q2 = 0 + -13.33333333
-8q + q2 = 0 + -13.33333333

Combine like terms: 0 + -13.33333333 = -13.33333333
-8q + q2 = -13.33333333

The q term is -8q.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8q + 16 + q2 = -13.33333333 + 16

Reorder the terms:
16 + -8q + q2 = -13.33333333 + 16

Combine like terms: -13.33333333 + 16 = 2.66666667
16 + -8q + q2 = 2.66666667

Factor a perfect square on the left side:
(q + -4)(q + -4) = 2.66666667

Calculate the square root of the right side: 1.632993163

Break this problem into two subproblems by setting 
(q + -4) equal to 1.632993163 and -1.632993163.


Subproblem 1
q + -4 = 1.632993163

Simplifying
q + -4 = 1.632993163

Reorder the terms:
-4 + q = 1.632993163

Solving
-4 + q = 1.632993163

Solving for variable 'q'.

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

Add '4' to each side of the equation.
-4 + 4 + q = 1.632993163 + 4

Combine like terms: -4 + 4 = 0
0 + q = 1.632993163 + 4
q = 1.632993163 + 4

Combine like terms: 1.632993163 + 4 = 5.632993163
q = 5.632993163

Simplifying
q = 5.632993163


Subproblem 2
q + -4 = -1.632993163

Simplifying
q + -4 = -1.632993163

Reorder the terms:
-4 + q = -1.632993163

Solving
-4 + q = -1.632993163

Solving for variable 'q'.

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

Add '4' to each side of the equation.
-4 + 4 + q = -1.632993163 + 4

Combine like terms: -4 + 4 = 0
0 + q = -1.632993163 + 4
q = -1.632993163 + 4

Combine like terms: -1.632993163 + 4 = 2.367006837
q = 2.367006837

Simplifying
q = 2.367006837


Solution
The solution to the problem is based on the solutions
from the subproblems.
q = {5.632993163, 2.367006837}

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