11+q=1/4q-2q

Solution for 11+q=1/4q-2q equation:

11+q=1/4q-2q

We move all terms to the left:

11+q-(1/4q-2q)=0


Domain of the equation: 4q-2q)!=0

q∈R


We add all the numbers together, and all the variables

q-(-2q+1/4q)+11=0

We get rid of parentheses

q+2q-1/4q+11=0

We multiply all the terms by the denominator


q*4q+2q*4q+11*4q-1=0

Wy multiply elements

4q^2+8q^2+44q-1=0

We add all the numbers together, and all the variables

12q^2+44q-1=0

a = 12; b = 44; c = -1;
Δ = b2-4ac
Δ = 442-4·12·(-1)
Δ = 1984
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$q_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$q_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{1984}=\sqrt{64*31}=\sqrt{64}*\sqrt{31}=8\sqrt{31}$
$q_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(44)-8\sqrt{31}}{2*12}=\frac{-44-8\sqrt{31}}{24}
$
$q_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(44)+8\sqrt{31}}{2*12}=\frac{-44+8\sqrt{31}}{24}
$