x-3/x-5=9

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

x-3/x-5=9

We move all terms to the left:

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


Domain of the equation: x!=0

x∈R


We add all the numbers together, and all the variables

x-3/x-14=0

We multiply all the terms by the denominator


x*x-14*x-3=0

We add all the numbers together, and all the variables

-14x+x*x-3=0

Wy multiply elements

x^2-14x-3=0

a = 1; b = -14; c = -3;
Δ = b2-4ac
Δ = -142-4·1·(-3)
Δ = 208
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:

$\sqrt{\Delta}=\sqrt{208}=\sqrt{16*13}=\sqrt{16}*\sqrt{13}=4\sqrt{13}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-14)-4\sqrt{13}}{2*1}=\frac{14-4\sqrt{13}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-14)+4\sqrt{13}}{2*1}=\frac{14+4\sqrt{13}}{2}
$