1/2+2/5r-1=1/5r+r

Solution for 1/2+2/5r-1=1/5r+r equation:

1/2+2/5r-1=1/5r+r

We move all terms to the left:

1/2+2/5r-1-(1/5r+r)=0


Domain of the equation: 5r!=0

r!=0/5

r!=0

r∈R



Domain of the equation: 5r+r)!=0

r∈R


We add all the numbers together, and all the variables

2/5r-(+r+1/5r)-1+1/2=0

We get rid of parentheses

2/5r-r-1/5r-1+1/2=0

We calculate fractions


-r+()/10r+5r/10r-1=0

We add all the numbers together, and all the variables

-1r+()/10r+5r/10r-1=0

We multiply all the terms by the denominator


-1r*10r+5r-1*10r+()=0

We add all the numbers together, and all the variables

5r-1r*10r-1*10r=0

Wy multiply elements

-10r^2+5r-10r=0

We add all the numbers together, and all the variables

-10r^2-5r=0

a = -10; b = -5; c = 0;
Δ = b2-4ac
Δ = -52-4·(-10)·0
Δ = 25
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{25}=5$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-5)-5}{2*-10}=\frac{0}{-20}
=0
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-5)+5}{2*-10}=\frac{10}{-20}
=-1/2
$