The common root of x^2+ax+b=0.
X^2+cx+d=0 is
Answers
for this problem we will use the property of the sum and product of roots of a quadratic
that is
if
α
&
β
are the roots of
p
x
2
+
q
x
+
r
=
0
then
α
β
=
−
q
p
α
β
=
r
p
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x
2
+
a
x
+
b
=
0
−
−
−
(
1
)
x
2
+
c
x
+
d
=
0
−
−
−
(
2
)
let the common root be
α
for eqn
(
1
)
α
+
α
=
−
a
⇒
α
=
−
a
2
&
α
2
=
b
for the eqn
(
2
)
let the second root be
β
then
α
+
β
=
−
c
α
β
=
d
⇒
β
=
d
α
∴
α
+
d
α
=
−
c
α
2
+
d
=
α
(
−
c
)
b
+
d
=
(
−
a
2
)
(
−
c
)
∴
2
(
b
+
d
)
=
a
c
as reqd.
Answer:
Step-by-step explanation:
Since, the eqn. x2+ax+b=0
has equal roots, its discriminant
must be zero.
∴a2−4b=0
or
b=a2*4.........(⋆).
Hence, the eqn. becomes,
x2+ax+a2*4=0.
a2-4b=0,
or
x=−a2.
This has to be a root of the second eqn. :
x2+cx+d=0
Substituting
x=−a2
in the second eqn., we have,
a2*4+c(a2)+d=0,
or, by (star), b−ac2+d=0.
This gives,
2(b+d)= ac
, as desired!