9+1/2q=5q-1

Solution for 9+1/2q=5q-1 equation:

9+1/2q=5q-1

We move all terms to the left:

9+1/2q-(5q-1)=0


Domain of the equation: 2q!=0

q!=0/2

q!=0

q∈R


We get rid of parentheses

1/2q-5q+1+9=0

We multiply all the terms by the denominator


-5q*2q+1*2q+9*2q+1=0

Wy multiply elements

-10q^2+2q+18q+1=0

We add all the numbers together, and all the variables

-10q^2+20q+1=0

a = -10; b = 20; c = +1;
Δ = b2-4ac
Δ = 202-4·(-10)·1
Δ = 440
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:

$\sqrt{\Delta}=\sqrt{440}=\sqrt{4*110}=\sqrt{4}*\sqrt{110}=2\sqrt{110}$
$q_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(20)-2\sqrt{110}}{2*-10}=\frac{-20-2\sqrt{110}}{-20}
$
$q_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(20)+2\sqrt{110}}{2*-10}=\frac{-20+2\sqrt{110}}{-20}
$