Question

If alpha and beta are the zeroes of a quadratic polynomial fx = kx2 + 4x +4 such that a2 + b2 = 24 find the value of k

Answer 1

Step-by-step explanation:

f(x)=kx^2+4x+4 a=k,b=4,c=4

a^2+b^2=24 alpha+beta=-4/k

(a+b)^2-2ab=24 alpha×beta=4/k

(-4/k)^2-2×4/k=24

16/k^2-8/k=24

16-8k/k^2=24

16-8k=24×k^2

8(2-k)=24k^2

2-k=24k^2/8

2-k=3k^2

0=3k^2+k-2

0=3k^2+3k-2k-2

0=3k(k+1)-2(k+1)

0=(3k-2)(k+1)

3k-2=0 k+1=0

k=2/3 k=-1

k=2/3 or -1

Answer 2

${\mathfrak{\purple{\underline{\underline{Solution:-}}}}}$

Given:

kx² + 4x +4 is a polynomial ; whose zeroes are a and ß.

Sum of zeroes:

a + ß = -4/k

Product of zeroes:

aß = 4/k

(a + b)² = a² + b² + 2ab

⇒ (-4/k)² = 24 + 2(4/k)

⇒ 16/k² = 24 + 8/k

⇒ 24k² + 8k - 16 = 0

⇒ 3k² + k - 2 = 0

⇒ 3k² + 3k - 2k -2 = 0

⇒ 3k(k + 1) - 2(k+1) = 0

⇒ (k + 1) (3k - 2) = 0

⇒ k = -1 or k = 2/3

Hence, the value of k is -1 or 2/3.