Question

If α and β are the zeroes of the polynomial k x2 + 4x + 4, show that α 2 β 2 = 24. Find the value of k.

Answer 1

k x²+4x+4
given ,α²+β²=24
α+β= -b/a= -4/k              αβ=c/a=4/k
squaring on both sides,
⇒α²+β²+2αβ=16/k²
substitute the values
⇒24+2(4/k)=16/k²
⇒24+8/k=16/k²
⇒24=16/k²-8/k
⇒24=(16-8k)/k²
⇒24k²=16-8k
⇒24k²+8k-16=0
⇒8(3k²+k-2)=0
⇒3k²+k-2=0
⇒3k²+3k-2k-2=0
⇒3k(k+1)-2(k+1)=0
⇒(3k-2)(k+1)=0
⇒3k-2=0        or                ⇒k+1=0
⇒k=2/3          or                ⇒k= -1