105=3Q^2-24Q

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Solution for 105=3Q^2-24Q equation: 


Simplifying
105 = 3Q2 + -24Q

Reorder the terms:
105 = -24Q + 3Q2

Solving
105 = -24Q + 3Q2

Solving for variable 'Q'.

Reorder the terms:
105 + 24Q + -3Q2 = -24Q + 24Q + 3Q2 + -3Q2

Combine like terms: -24Q + 24Q = 0
105 + 24Q + -3Q2 = 0 + 3Q2 + -3Q2
105 + 24Q + -3Q2 = 3Q2 + -3Q2

Combine like terms: 3Q2 + -3Q2 = 0
105 + 24Q + -3Q2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(35 + 8Q + -1Q2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(35 + 8Q + -1Q2)' equal to zero and attempt to solve:

Simplifying
35 + 8Q + -1Q2 = 0

Solving
35 + 8Q + -1Q2 = 0

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

Divide each side by '-1'.
-35 + -8Q + Q2 = 0

Move the constant term to the right:

Add '35' to each side of the equation.
-35 + -8Q + 35 + Q2 = 0 + 35

Reorder the terms:
-35 + 35 + -8Q + Q2 = 0 + 35

Combine like terms: -35 + 35 = 0
0 + -8Q + Q2 = 0 + 35
-8Q + Q2 = 0 + 35

Combine like terms: 0 + 35 = 35
-8Q + Q2 = 35

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 = 35 + 16

Reorder the terms:
16 + -8Q + Q2 = 35 + 16

Combine like terms: 35 + 16 = 51
16 + -8Q + Q2 = 51

Factor a perfect square on the left side:
(Q + -4)(Q + -4) = 51

Calculate the square root of the right side: 7.141428429

Break this problem into two subproblems by setting 
(Q + -4) equal to 7.141428429 and -7.141428429.


Subproblem 1
Q + -4 = 7.141428429

Simplifying
Q + -4 = 7.141428429

Reorder the terms:
-4 + Q = 7.141428429

Solving
-4 + Q = 7.141428429

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 = 7.141428429 + 4

Combine like terms: -4 + 4 = 0
0 + Q = 7.141428429 + 4
Q = 7.141428429 + 4

Combine like terms: 7.141428429 + 4 = 11.141428429
Q = 11.141428429

Simplifying
Q = 11.141428429


Subproblem 2
Q + -4 = -7.141428429

Simplifying
Q + -4 = -7.141428429

Reorder the terms:
-4 + Q = -7.141428429

Solving
-4 + Q = -7.141428429

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 = -7.141428429 + 4

Combine like terms: -4 + 4 = 0
0 + Q = -7.141428429 + 4
Q = -7.141428429 + 4

Combine like terms: -7.141428429 + 4 = -3.141428429
Q = -3.141428429

Simplifying
Q = -3.141428429


Solution
The solution to the problem is based on the solutions
from the subproblems.
Q = {11.141428429, -3.141428429}
Solution
Q = {11.141428429, -3.141428429}

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