x-30=1/2x-22

Solution for x-30=1/2x-22 equation:

x-30=1/2x-22

We move all terms to the left:

x-30-(1/2x-22)=0


Domain of the equation: 2x-22)!=0

x∈R


We get rid of parentheses

x-1/2x+22-30=0

We multiply all the terms by the denominator


x*2x+22*2x-30*2x-1=0

Wy multiply elements

2x^2+44x-60x-1=0

We add all the numbers together, and all the variables

2x^2-16x-1=0

a = 2; b = -16; c = -1;
Δ = b2-4ac
Δ = -162-4·2·(-1)
Δ = 264
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:

$\sqrt{\Delta}=\sqrt{264}=\sqrt{4*66}=\sqrt{4}*\sqrt{66}=2\sqrt{66}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-16)-2\sqrt{66}}{2*2}=\frac{16-2\sqrt{66}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-16)+2\sqrt{66}}{2*2}=\frac{16+2\sqrt{66}}{4}
$