(24Q-Q^2)-(20+8Q+Q^2)=0

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Solution for (24Q-Q^2)-(20+8Q+Q^2)=0 equation: 


Simplifying
(24Q + -1Q2) + -1(20 + 8Q + Q2) = 0

Remove parenthesis around (24Q + -1Q2)
24Q + -1Q2 + -1(20 + 8Q + Q2) = 0
24Q + -1Q2 + (20 * -1 + 8Q * -1 + Q2 * -1) = 0
24Q + -1Q2 + (-20 + -8Q + -1Q2) = 0

Reorder the terms:
-20 + 24Q + -8Q + -1Q2 + -1Q2 = 0

Combine like terms: 24Q + -8Q = 16Q
-20 + 16Q + -1Q2 + -1Q2 = 0

Combine like terms: -1Q2 + -1Q2 = -2Q2
-20 + 16Q + -2Q2 = 0

Solving
-20 + 16Q + -2Q2 = 0

Solving for variable 'Q'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-10 + 8Q + -1Q2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-10 + 8Q + -1Q2)' equal to zero and attempt to solve:

Simplifying
-10 + 8Q + -1Q2 = 0

Solving
-10 + 8Q + -1Q2 = 0

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

Divide each side by '-1'.
10 + -8Q + Q2 = 0

Move the constant term to the right:

Add '-10' to each side of the equation.
10 + -8Q + -10 + Q2 = 0 + -10

Reorder the terms:
10 + -10 + -8Q + Q2 = 0 + -10

Combine like terms: 10 + -10 = 0
0 + -8Q + Q2 = 0 + -10
-8Q + Q2 = 0 + -10

Combine like terms: 0 + -10 = -10
-8Q + Q2 = -10

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 = -10 + 16

Reorder the terms:
16 + -8Q + Q2 = -10 + 16

Combine like terms: -10 + 16 = 6
16 + -8Q + Q2 = 6

Factor a perfect square on the left side:
(Q + -4)(Q + -4) = 6

Calculate the square root of the right side: 2.449489743

Break this problem into two subproblems by setting 
(Q + -4) equal to 2.449489743 and -2.449489743.


Subproblem 1
Q + -4 = 2.449489743

Simplifying
Q + -4 = 2.449489743

Reorder the terms:
-4 + Q = 2.449489743

Solving
-4 + Q = 2.449489743

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 = 2.449489743 + 4

Combine like terms: -4 + 4 = 0
0 + Q = 2.449489743 + 4
Q = 2.449489743 + 4

Combine like terms: 2.449489743 + 4 = 6.449489743
Q = 6.449489743

Simplifying
Q = 6.449489743


Subproblem 2
Q + -4 = -2.449489743

Simplifying
Q + -4 = -2.449489743

Reorder the terms:
-4 + Q = -2.449489743

Solving
-4 + Q = -2.449489743

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 = -2.449489743 + 4

Combine like terms: -4 + 4 = 0
0 + Q = -2.449489743 + 4
Q = -2.449489743 + 4

Combine like terms: -2.449489743 + 4 = 1.550510257
Q = 1.550510257

Simplifying
Q = 1.550510257


Solution
The solution to the problem is based on the solutions
from the subproblems.
Q = {6.449489743, 1.550510257}
Solution
Q = {6.449489743, 1.550510257}

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