w(w+1)=90

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

Simplifying
w(w + 1) = 90

Reorder the terms:
w(1 + w) = 90
(1 * w + w * w) = 90
(1w + w2) = 90

Solving
1w + w2 = 90

Solving for variable 'w'.

Reorder the terms:
-90 + 1w + w2 = 90 + -90

Combine like terms: 90 + -90 = 0
-90 + 1w + w2 = 0

Factor a trinomial.
(-10 + -1w)(9 + -1w) = 0

Subproblem 1
Set the factor '(-10 + -1w)' equal to zero and attempt to solve:

Simplifying
-10 + -1w = 0

Solving
-10 + -1w = 0

Move all terms containing w to the left, all other terms to the right.

Add '10' to each side of the equation.
-10 + 10 + -1w = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -1w = 0 + 10
-1w = 0 + 10

Combine like terms: 0 + 10 = 10
-1w = 10

Divide each side by '-1'.
w = -10

Simplifying
w = -10
Subproblem 2
Set the factor '(9 + -1w)' equal to zero and attempt to solve:

Simplifying
9 + -1w = 0

Solving
9 + -1w = 0

Move all terms containing w to the left, all other terms to the right.

Add '-9' to each side of the equation.
9 + -9 + -1w = 0 + -9

Combine like terms: 9 + -9 = 0
0 + -1w = 0 + -9
-1w = 0 + -9

Combine like terms: 0 + -9 = -9
-1w = -9

Divide each side by '-1'.
w = 9

Simplifying
w = 9
Solution
w = {-10, 9}