3/(1-4x)=x-2

Solution for 3/(1-4x)=x-2 equation:

3/(1-4x)=x-2

We move all terms to the left:

3/(1-4x)-(x-2)=0


Domain of the equation: (1-4x)!=0

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

-4x!=-1

x!=-1/-4

x!=1/4

x∈R


We add all the numbers together, and all the variables

3/(-4x+1)-(x-2)=0

We get rid of parentheses

3/(-4x+1)-x+2=0

We multiply all the terms by the denominator


-x*(-4x+1)+2*(-4x+1)+3=0

We multiply parentheses

4x^2-x-8x+2+3=0

We add all the numbers together, and all the variables

4x^2-9x+5=0

a = 4; b = -9; c = +5;
Δ = b2-4ac
Δ = -92-4·4·5
Δ = 1
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{1}=1$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-9)-1}{2*4}=\frac{8}{8}
=1
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-9)+1}{2*4}=\frac{10}{8}
=1+1/4
$