x-5/x-4=1

Solution for x-5/x-4=1 equation:

x-5/x-4=1

We move all terms to the left:

x-5/x-4-(1)=0


Domain of the equation: x!=0

x∈R


We add all the numbers together, and all the variables

x-5/x-5=0

We multiply all the terms by the denominator


x*x-5*x-5=0

We add all the numbers together, and all the variables

-5x+x*x-5=0

Wy multiply elements

x^2-5x-5=0

a = 1; b = -5; c = -5;
Δ = b2-4ac
Δ = -52-4·1·(-5)
Δ = 45
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:

$\sqrt{\Delta}=\sqrt{45}=\sqrt{9*5}=\sqrt{9}*\sqrt{5}=3\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-5)-3\sqrt{5}}{2*1}=\frac{5-3\sqrt{5}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-5)+3\sqrt{5}}{2*1}=\frac{5+3\sqrt{5}}{2}
$