(4x+1)/(3x-5)=x

Solution for (4x+1)/(3x-5)=x equation:

(4x+1)/(3x-5)=x

We move all terms to the left:

(4x+1)/(3x-5)-(x)=0


Domain of the equation: (3x-5)!=0

We move all terms containing x to the left, all other terms to the right

3x!=5

x!=5/3

x!=1+2/3

x∈R


We add all the numbers together, and all the variables

-1x+(4x+1)/(3x-5)=0

We multiply all the terms by the denominator


-1x*(3x-5)+(4x+1)=0

We multiply parentheses

-3x^2+5x+(4x+1)=0

We get rid of parentheses

-3x^2+5x+4x+1=0

We add all the numbers together, and all the variables

-3x^2+9x+1=0

a = -3; b = 9; c = +1;
Δ = b2-4ac
Δ = 92-4·(-3)·1
Δ = 93
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(9)-\sqrt{93}}{2*-3}=\frac{-9-\sqrt{93}}{-6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(9)+\sqrt{93}}{2*-3}=\frac{-9+\sqrt{93}}{-6}
$