r-9=6/r-5=9

Solution for r-9=6/r-5=9 equation:

r-9=6/r-5=9

We move all terms to the left:

r-9-(6/r-5)=0


Domain of the equation: r-5)!=0

r∈R


We get rid of parentheses

r-6/r+5-9=0

We multiply all the terms by the denominator


r*r+5*r-9*r-6=0

We add all the numbers together, and all the variables

-4r+r*r-6=0

Wy multiply elements

r^2-4r-6=0

a = 1; b = -4; c = -6;
Δ = b2-4ac
Δ = -42-4·1·(-6)
Δ = 40
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:

$\sqrt{\Delta}=\sqrt{40}=\sqrt{4*10}=\sqrt{4}*\sqrt{10}=2\sqrt{10}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{10}}{2*1}=\frac{4-2\sqrt{10}}{2}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{10}}{2*1}=\frac{4+2\sqrt{10}}{2}
$