# Solve for n n(n+1)=90 n(n+1)=90
Simplify n(n+1).
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
Factor n2+n-90 using the AC method.
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 and solve.
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 and solve.
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

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