2/(4x-1)=x+7

Solution for 2/(4x-1)=x+7 equation:

2/(4x-1)=x+7

We move all terms to the left:

2/(4x-1)-(x+7)=0


Domain of the equation: (4x-1)!=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 get rid of parentheses

2/(4x-1)-x-7=0

We multiply all the terms by the denominator


-x*(4x-1)-7*(4x-1)+2=0

We multiply parentheses

-4x^2+1x-28x+7+2=0

We add all the numbers together, and all the variables

-4x^2-27x+9=0

a = -4; b = -27; c = +9;
Δ = b2-4ac
Δ = -272-4·(-4)·9
Δ = 873
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:

$\sqrt{\Delta}=\sqrt{873}=\sqrt{9*97}=\sqrt{9}*\sqrt{97}=3\sqrt{97}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-27)-3\sqrt{97}}{2*-4}=\frac{27-3\sqrt{97}}{-8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-27)+3\sqrt{97}}{2*-4}=\frac{27+3\sqrt{97}}{-8}
$