1/2q+3=2q+9

Solution for 1/2q+3=2q+9 equation:

1/2q+3=2q+9

We move all terms to the left:

1/2q+3-(2q+9)=0


Domain of the equation: 2q!=0

q!=0/2

q!=0

q∈R


We get rid of parentheses

1/2q-2q-9+3=0

We multiply all the terms by the denominator


-2q*2q-9*2q+3*2q+1=0

Wy multiply elements

-4q^2-18q+6q+1=0

We add all the numbers together, and all the variables

-4q^2-12q+1=0

a = -4; b = -12; c = +1;
Δ = b2-4ac
Δ = -122-4·(-4)·1
Δ = 160
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:

$\sqrt{\Delta}=\sqrt{160}=\sqrt{16*10}=\sqrt{16}*\sqrt{10}=4\sqrt{10}$
$q_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-4\sqrt{10}}{2*-4}=\frac{12-4\sqrt{10}}{-8}
$
$q_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+4\sqrt{10}}{2*-4}=\frac{12+4\sqrt{10}}{-8}
$