Verified Answer :

1 Simplify
n
(
n
−
1
)
(
n
−
2
)
×
1
n(n−1)(n−2)×1 to
n
(
n
−
1
)
(
n
−
2
)
n(n−1)(n−2)
n
=
n
(
n
−
1
)
(
n
−
2
)
n=n(n−1)(n−2)

2 Expand
n
=
n
3
−
2
n
2
−
n
2
+
2
n
n=n
​3
​​ −2n
​2
​​ −n
​2
​​ +2n

3 Simplify
n
3
−
2
n
2
−
n
2
+
2
n
n
​3
​​ −2n
​2
​​ −n
​2
​​ +2n to
n
3
−
3
n
2
+
2
n
n
​3
​​ −3n
​2
​​ +2n
n
=
n
3
−
3
n
2
+
2
n
n=n
​3
​​ −3n
​2
​​ +2n

4 Move all terms to one side
n
−
n
3
+
3
n
2
−
2
n
=
0
n−n
​3
​​ +3n
​2
​​ −2n=0

5 Simplify
n
−
n
3
+
3
n
2
−
2
n
n−n
​3
​​ +3n
​2
​​ −2n to
−
n
−
n
3
+
3
n
2
−n−n
​3
​​ +3n
​2
​​
−
n
−
n
3
+
3
n
2
=
0
−n−n
​3
​​ +3n
​2
​​ =0

6 Factor out the common term
n
n
−
n
(
1
+
n
2
−
3
n
)
=
0
−n(1+n
​2
​​ −3n)=0

7 Multiply both sides by
−
1
−1
n
(
1
+
n
2
−
3
n
)
=
0
n(1+n
​2
​​ −3n)=0

8 Solve for
n
n
n
=
0
n=0

9 Apply the Quadratic FormulaHow?
n
=
3
+
5
2
,
3
−
5
2
n=
​2
​
​3+√
​5
​
​​
​​ ,
​2
​
​3−√
​5
​
​​
​​

10 Collect all solutions from the previous steps
n
=
0
,
3
+
5
2
,
3
−
5
2
n=0,
​2
​
​3+√
​5
​
​​
​​ ,
​2
​
​3−√
​5
​
​​
​​

Done

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