1/2r+6=33-2r

Solution for 1/2r+6=33-2r equation:

1/2r+6=33-2r

We move all terms to the left:

1/2r+6-(33-2r)=0


Domain of the equation: 2r!=0

r!=0/2

r!=0

r∈R


We add all the numbers together, and all the variables

1/2r-(-2r+33)+6=0

We get rid of parentheses

1/2r+2r-33+6=0

We multiply all the terms by the denominator


2r*2r-33*2r+6*2r+1=0

Wy multiply elements

4r^2-66r+12r+1=0

We add all the numbers together, and all the variables

4r^2-54r+1=0

a = 4; b = -54; c = +1;
Δ = b2-4ac
Δ = -542-4·4·1
Δ = 2900
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{2900}=\sqrt{100*29}=\sqrt{100}*\sqrt{29}=10\sqrt{29}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-54)-10\sqrt{29}}{2*4}=\frac{54-10\sqrt{29}}{8}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-54)+10\sqrt{29}}{2*4}=\frac{54+10\sqrt{29}}{8}
$