1/2x-2=4x-23

Solution for 1/2x-2=4x-23 equation:

1/2x-2=4x-23

We move all terms to the left:

1/2x-2-(4x-23)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-4x+23-2=0

We multiply all the terms by the denominator


-4x*2x+23*2x-2*2x+1=0

Wy multiply elements

-8x^2+46x-4x+1=0

We add all the numbers together, and all the variables

-8x^2+42x+1=0

a = -8; b = 42; c = +1;
Δ = b2-4ac
Δ = 422-4·(-8)·1
Δ = 1796
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:

$\sqrt{\Delta}=\sqrt{1796}=\sqrt{4*449}=\sqrt{4}*\sqrt{449}=2\sqrt{449}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(42)-2\sqrt{449}}{2*-8}=\frac{-42-2\sqrt{449}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(42)+2\sqrt{449}}{2*-8}=\frac{-42+2\sqrt{449}}{-16}
$