k(k-1)+(k-1)=5k+k

Solution for k(k-1)+(k-1)=5k+k equation:

k(k-1)+(k-1)=5k+k

We move all terms to the left:

k(k-1)+(k-1)-(5k+k)=0

We add all the numbers together, and all the variables

k(k-1)+(k-1)-(+6k)=0

We multiply parentheses

k^2-1k+(k-1)-(+6k)=0

We get rid of parentheses

k^2-1k+k-6k-1=0

We add all the numbers together, and all the variables

k^2-6k-1=0

a = 1; b = -6; c = -1;
Δ = b2-4ac
Δ = -62-4·1·(-1)
Δ = 40
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{40}=\sqrt{4*10}=\sqrt{4}*\sqrt{10}=2\sqrt{10}$
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{10}}{2*1}=\frac{6-2\sqrt{10}}{2}
$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{10}}{2*1}=\frac{6+2\sqrt{10}}{2}
$