1/2*x+5=10+x

Solution for 1/2*x+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^2-10x+1=0

a = -2; b = -10; c = +1;
Δ = b2-4ac
Δ = -102-4·(-2)·1
Δ = 108
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:

$\sqrt{\Delta}=\sqrt{108}=\sqrt{36*3}=\sqrt{36}*\sqrt{3}=6\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-10)-6\sqrt{3}}{2*-2}=\frac{10-6\sqrt{3}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-10)+6\sqrt{3}}{2*-2}=\frac{10+6\sqrt{3}}{-4}
$