If x
2
−3x+1=0, then (x
3
+x
−3
) is equal to:

Given equation:
x
2
−3x+1=0
Dividing both sides by x (x

=0):

x−3+
x
1
​
=0⟹x+
x
1
​
=3
We need to find the value of x
3
+x
−3
=x
3
+
x
3

1
​
.

Using the algebraic identity:

x
3
+
x
3

1
​
=(x+
x
1
​
)
3
−3(x+
x
1
​
)
Substitute the value x+
x
1
​
=3:

x
3
+
x
3

1
​
=(3)
3
−3(3)=27−9=18