n(n+1)=90

Apply the distributive property.

n⋅n+n⋅1=90

Simplify the expression.

Multiply n by n.

n2+n⋅1=90

Multiply n by 1.

n2+n=90

n2+n=90

n2+n=90

Move 90 to the left side of the equation by subtracting it from both sides.

n2+n-90=0

Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is -90 and whose sum is 1.

-9,10

Write the factored form using these integers.

(n-9)(n+10)=0

(n-9)(n+10)=0

If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.

n-9=0

n+10=0

Set the first factor equal to 0.

n-9=0

Add 9 to both sides of the equation.

n=9

n=9

Set the next factor equal to 0.

n+10=0

Subtract 10 from both sides of the equation.

n=-10

n=-10

The final solution is all the values that make (n-9)(n+10)=0 true.

n=9,-10

Solve for n n(n+1)=90