2/3r+r=10

Solution for 2/3r+r=10 equation:

2/3r+r=10

We move all terms to the left:

2/3r+r-(10)=0


Domain of the equation: 3r!=0

r!=0/3

r!=0

r∈R


We add all the numbers together, and all the variables

r+2/3r-10=0

We multiply all the terms by the denominator


r*3r-10*3r+2=0

Wy multiply elements

3r^2-30r+2=0

a = 3; b = -30; c = +2;
Δ = b2-4ac
Δ = -302-4·3·2
Δ = 876
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{876}=\sqrt{4*219}=\sqrt{4}*\sqrt{219}=2\sqrt{219}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-30)-2\sqrt{219}}{2*3}=\frac{30-2\sqrt{219}}{6}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-30)+2\sqrt{219}}{2*3}=\frac{30+2\sqrt{219}}{6}
$