k(k+1)=5.25

Solution for k(k+1)=5.25 equation:

k(k+1)=5.25

We move all terms to the left:

k(k+1)-(5.25)=0

We add all the numbers together, and all the variables

k(k+1)-5.25=0

We multiply parentheses

k^2+k-5.25=0

a = 1; b = 1; c = -5.25;
Δ = b2-4ac
Δ = 12-4·1·(-5.25)
Δ = 22
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-\sqrt{22}}{2*1}=\frac{-1-\sqrt{22}}{2}
$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+\sqrt{22}}{2*1}=\frac{-1+\sqrt{22}}{2}
$