-1/2x+5=x-10

Solution for -1/2x+5=x-10 equation:

-1/2x+5=x-10

We move all terms to the left:

-1/2x+5-(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+5=0

We multiply all the terms by the denominator


-x*2x+10*2x+5*2x-1=0

Wy multiply elements

-2x^2+20x+10x-1=0

We add all the numbers together, and all the variables

-2x^2+30x-1=0

a = -2; b = 30; c = -1;
Δ = b2-4ac
Δ = 302-4·(-2)·(-1)
Δ = 892
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:

$\sqrt{\Delta}=\sqrt{892}=\sqrt{4*223}=\sqrt{4}*\sqrt{223}=2\sqrt{223}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(30)-2\sqrt{223}}{2*-2}=\frac{-30-2\sqrt{223}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(30)+2\sqrt{223}}{2*-2}=\frac{-30+2\sqrt{223}}{-4}
$