1/2x-1=7x-27

Solution for 1/2x-1=7x-27 equation:

1/2x-1=7x-27

We move all terms to the left:

1/2x-1-(7x-27)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-7x+27-1=0

We multiply all the terms by the denominator


-7x*2x+27*2x-1*2x+1=0

Wy multiply elements

-14x^2+54x-2x+1=0

We add all the numbers together, and all the variables

-14x^2+52x+1=0

a = -14; b = 52; c = +1;
Δ = b2-4ac
Δ = 522-4·(-14)·1
Δ = 2760
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:

$\sqrt{\Delta}=\sqrt{2760}=\sqrt{4*690}=\sqrt{4}*\sqrt{690}=2\sqrt{690}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(52)-2\sqrt{690}}{2*-14}=\frac{-52-2\sqrt{690}}{-28}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(52)+2\sqrt{690}}{2*-14}=\frac{-52+2\sqrt{690}}{-28}
$