Doubt by Zoha
Solution :
A quadratic polynomial can only be expressed as a perfect square if and only if it has two equal factors or it has two equal zeroes.
Let us understand this with an example.
Let us consider a quadratic polynomial
p(x)=x²+4x+4
p(x)=x²+(2+2)x+4
p(x)=x²+2x+2x+4
p(x)=x(x+2)+2(x+2)
p(x)=(x+2)(x+2)
p(x)=(x+2)²
Now, lets find out the zeroes of this polynomial
p(x)=0
(x+2)(x+2)=0
x+2=0 or x+2=0
x=-2 or x=-2
x=-2,-2
Now, we could also express it as —
Every quadratic equation can be expressed as a perfect square if it has two real and equal zeroes i.e. its discriminate (D) must be equal to zero.
(4-k)x²+(2k+4)x+(8k+1)=0
a=(4-k)
b=(2k+4)
c=8k+1
D=b²-4ac
D=(2k+4)²-4(4-k)(8k+1)
D=[2(k+2)]²-4[4(8k+1)-k(8k+1)]
D=4(k+2)²-4[32k+4-8k²-k]
0=4[k²+4+4k-32k-4+8k²+k]
[∵ Roots are real and equal]
[∵ Roots are real and equal]
0/4 = 9k²-27k
0=9k(k-3)
9k=0 or (k-3)=0
k=0/9 or k=3
k=0 or k=3
Hence, the required value of k must be 0 or 3.