Question

for what values of k the expression kx^2 + (k+1)x + 2 will be a perfect square of linear polynomial

Answer 1

Let ax + b is a linear polynomial .
A/C to question,
(ax + b)² = kx² + (k + 1)x + 2
a²x² + 2abx + b² = kx² + (k + 1)x + 2
compare both sides,
a² = k , 2ab = ( k + 1) and b² = 2
∵ b² = 2
Taking square root both sides,
b = ±√2
2ab = (k + 1) , put b = ±√2
±2√2a = (k + 1)
Now squaring both sides,
8a² = k² + 2k + 1 , put a² = k
8k = k² + 2k + 1
k² - 6k + 1 = 0
k = {6 ± 4√2}/2 = 3 ± 2√2

Hence, k = (3 ± 2√2)

Answer 2

Hey! ! !

Solution :-

☆ For perfect square :-

☆ b^2 - 4ac=0

=> 4(k+1)^2 - 4.k^2.4=0

=> (k+1)^2 - 4k^2 =0

=> k^2+2k+1–4k^2=0

=> 3k^2–2k-1=0

=> 3k^2–3k+k-1=0

=> 3k(k-1)+1(k-1)=0

=> (k-1)(3k+1)=0

:- k= 1 :- = -1/3 Answer.

☆ ☆ ☆ Hop its helpful ☆ ☆ ☆

☆ Regards :- ♡♡《 Nitish kr singh 》♡♡