90=k(1+k)

Solution for 90=k(1+k) equation:

90=k(1+k)

We move all terms to the left:

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

We add all the numbers together, and all the variables

-(k(k+1))+90=0


We calculate terms in parentheses: -(k(k+1)), so:

k(k+1)

We multiply parentheses

k^2+k

Back to the equation:

-(k^2+k)


We get rid of parentheses

-k^2-k+90=0

We add all the numbers together, and all the variables

-1k^2-1k+90=0

a = -1; b = -1; c = +90;
Δ = b2-4ac
Δ = -12-4·(-1)·90
Δ = 361
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{361}=19$
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-19}{2*-1}=\frac{-18}{-2}
=+9
$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+19}{2*-1}=\frac{20}{-2}
=-10
$