1/2x-5=10+x

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

1/2x-5=10+x

We move all terms to the left:

1/2x-5-(10+x)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

1/2x-(x+10)-5=0

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 = 908
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:

$\sqrt{\Delta}=\sqrt{908}=\sqrt{4*227}=\sqrt{4}*\sqrt{227}=2\sqrt{227}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-30)-2\sqrt{227}}{2*-2}=\frac{30-2\sqrt{227}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-30)+2\sqrt{227}}{2*-2}=\frac{30+2\sqrt{227}}{-4}
$