find the value of 'p' so that the equation 4x^2-8px+9=0 has roots whose difference is 4
Answers
Therefore, for x=a
4x^2 -8px +9 = 4a^2 -8pa +9 = 0
And for x=a+4,
4x^2 -8px +9 = 4(a+4)^2-8p(a+4)+9=0
Therefore,
4a^2 -8pa +9 = 4(a^2 +16 +8a) -8ap -32p +9 = 0
4a^2 -8pa +9 = 4a^2 +64 +32a -8ap -32p +9 = 0
Answer:
Step-by-step explanation:
We know that if m and n are the roots of a quadratic equation ax
2
+bx+c=0, the sum of the roots is m+n=−
a
b
and the product of the roots is mn=
a
c
.
Let m and n be the roots of the given quadratic equation 4x
2
−8px+9=0. It is given that the difference of the roots is 4, therefore,
m−n=4........(1)
The equation 4x
2
−8px+9=0 is in the form ax
2
+bx+c=0 where a=4,b=−8q and c=9.
The sum of the roots is:
m+n=−
a
b
=−
4
(−8q)
=2q....(2)
The product of the roots is
a
c
that is:
mn=
a
c
=
4
9
......(3)
Now, we know the identity (m+n)
2
=(m−n)
2
+4mn, therefore, using equations 1,2 and 3, we have
(m+n)
2
=(m−n)
2
+4mn
⇒(2p)
2
=4
2
+(4×
4
9
)
⇒4p
2
=16+9
⇒4p
2
=25
⇒p
2
=
4
25
⇒p=±
4
25
⇒p=±
2
5
Hence, the value of p=±
2
5
.