Question

if x= -2 is root of equation 3x^2+7x+p=0,find the value of k so that the roots of equation x^2+k(4x+k-1)+p=0are equal.

Answer 1

x = -2
Then,
P(-2) = 3(-2)^ + 7(-2) + p =0
= 3*4-14+p=0
=12-14+p=0
=-2+p=0
p = 2. -----------[1]

x^2 + k(4x+k-1)+2=0----------[From 1]
x^2 + 4kx + k(k-1) + 2 = 0
a=1, b=4k, c=k(k-1)+2


⇒ D = b^2 – 4ac = 0

⇒ (4K)^2 – 4 [1] [ K(K – 1) + 2 ] = 0

16K^2 – 4 [ K^2 – K + 2 ] = 0

16K^2 – 4K^2 + 4K – 8 = 0

12K^2 + 4K – 8 = 0

3K^2 + K – 2 = 0

3K^2 + 3K – 2K – 2 = 0

⇒ 3K ( K +1 ) – 2 ( K + 1) = 0

⇒ ( K +1 ) ( 3K – 2) = 0

Therefore, K = -1 or K = 2/3

Answer 2

Answer:

x=-2

p[x]=3 x²+7 x+p=0

p[-2]=3[-2]²+7[-2]+p=0

       =12-14+p=0

       =-2+p=0

      p=2

x²+k[4x+k-1]+p=0

x²+4kx+k²-k+2=0    [put p=2]

D=b²-4ac=0

a=1  , b=4k  ,c=k²-k+2

D=[4k]²-4(1)(k²-k+2)=0

 =16k²-4k²+4k-8=0

 =12k²+4k-8=0

 =3k²+k-2=0  (taking 4 common)

 =3k²+3k-2k-2=0    (splitting middle term)

 =3k(k+1) -2(k+1)=0

 =(k+1)(3k-2)=0

 ∴k=-1 or 2/3

 

Step-by-step explanation: