Magic of Discriminant
Consider x 2 + b x + c = 0 such that b 2 − 4 c + 3 = 0 . If r and s are the roots of x 2 + b x + c = 0 , find ( r − s ) ( s − r ) .
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
2 solutions
Alter: Discriminant of given equation, b 2 − 4 c =-3 It is clear that roots of the given equation will be of the form 2 − b + q and 2 − b − q Where q is the discriminant. Now, ( r − s ) × ( s − r ) = 4 2 q × − 2 q = 4 − 4 q =-q =3
actually the defination of discriminant for a monicc polynomial is the product of the squareof the differences of roots taken symmetrically
⇒
x
2
+
b
x
+
c
=
0
Root are
r
and
s
.
By
Vieta's
formula.
●
r
+
s
=
−
b
●
r
s
=
c
⇒
(
r
−
s
)
(
s
−
r
)
r
s
−
r
2
−
s
2
+
r
s
2
r
s
−
(
r
2
+
s
2
)
2
r
s
−
[
(
r
+
s
)
2
−
2
r
s
]
2
c
−
[
b
2
−
2
c
]
......
(
1
)
⇒
b
2
−
4
c
+
3
=
0
b
2
−
2
c
−
2
c
+
3
=
0
b
2
−
2
c
=
2
c
−
3
Putting in
(
1
)
.
2
c
−
[
2
c
−
3
]
2
c
−
2
c
+
3
3
Easiest way is to take b=c=1