3x-9=9/x+5

Solution for 3x-9=9/x+5 equation:

3x-9=9/x+5

We move all terms to the left:

3x-9-(9/x+5)=0


Domain of the equation: x+5)!=0

x∈R


We get rid of parentheses

3x-9/x-5-9=0

We multiply all the terms by the denominator


3x*x-5*x-9*x-9=0

We add all the numbers together, and all the variables

-14x+3x*x-9=0

Wy multiply elements

3x^2-14x-9=0

a = 3; b = -14; c = -9;
Δ = b2-4ac
Δ = -142-4·3·(-9)
Δ = 304
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{304}=\sqrt{16*19}=\sqrt{16}*\sqrt{19}=4\sqrt{19}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-14)-4\sqrt{19}}{2*3}=\frac{14-4\sqrt{19}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-14)+4\sqrt{19}}{2*3}=\frac{14+4\sqrt{19}}{6}
$