3r^2-4r-13=0

Solution for 3r^2-4r-13=0 equation:

3r^2-4r-13=0

a = 3; b = -4; c = -13;
Δ = b2-4ac
Δ = -42-4·3·(-13)
Δ = 172
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:

$\sqrt{\Delta}=\sqrt{172}=\sqrt{4*43}=\sqrt{4}*\sqrt{43}=2\sqrt{43}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{43}}{2*3}=\frac{4-2\sqrt{43}}{6}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{43}}{2*3}=\frac{4+2\sqrt{43}}{6}
$