1/2x-22=x-10

Solution for 1/2x-22=x-10 equation:

1/2x-22=x-10

We move all terms to the left:

1/2x-22-(x-10)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-x+10-22=0

We multiply all the terms by the denominator


-x*2x+10*2x-22*2x+1=0

Wy multiply elements

-2x^2+20x-44x+1=0

We add all the numbers together, and all the variables

-2x^2-24x+1=0

a = -2; b = -24; c = +1;
Δ = b2-4ac
Δ = -242-4·(-2)·1
Δ = 584
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:

$\sqrt{\Delta}=\sqrt{584}=\sqrt{4*146}=\sqrt{4}*\sqrt{146}=2\sqrt{146}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-24)-2\sqrt{146}}{2*-2}=\frac{24-2\sqrt{146}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-24)+2\sqrt{146}}{2*-2}=\frac{24+2\sqrt{146}}{-4}
$