1/2x-5=4x-61

Solution for 1/2x-5=4x-61 equation:

1/2x-5=4x-61

We move all terms to the left:

1/2x-5-(4x-61)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-4x+61-5=0

We multiply all the terms by the denominator


-4x*2x+61*2x-5*2x+1=0

Wy multiply elements

-8x^2+122x-10x+1=0

We add all the numbers together, and all the variables

-8x^2+112x+1=0

a = -8; b = 112; c = +1;
Δ = b2-4ac
Δ = 1122-4·(-8)·1
Δ = 12576
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:

$\sqrt{\Delta}=\sqrt{12576}=\sqrt{16*786}=\sqrt{16}*\sqrt{786}=4\sqrt{786}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(112)-4\sqrt{786}}{2*-8}=\frac{-112-4\sqrt{786}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(112)+4\sqrt{786}}{2*-8}=\frac{-112+4\sqrt{786}}{-16}
$