Question

If alpha and beta are the zeroes of
quadratic polynomial
f (x) = K(x) = Kx² + 4x +4 such that
alpha² + Beta² =24, find the
values of K.​

Answer 1

Step-by-step explanation:

α,β roots of f(x)=kx

2

+4x+4

Given α

2

+β

2

=24

We know α+β=

a

−b

=

k

−4

αβ=

a

c

=

k

4

(α+β)

2

=α

2

+β

2

+2αβ

(

k

−4

)

2

=24+2(

k

4

)

k

2

4

2

=24+2(

k

4

)

16=24k

2

+8k

2=3k

2

+k

0=3k

2

+k−2

0=3k

2

+3k−2k−2

0=3k(k+1)−2(k+1)

0=(k+1)(3k−2)

∴k=−1,

3

2

Answer 2

$\huge\pink{\mid{\fbox{\tt{Solution}}\mid}}$

a,β roots of f(x)= kx²+4x+4

Given:-

a²+β²=24

${We\;know\;a+β\implies\frac{-b}{a}=\frac{-4}{k}}$

$\implies{aβ=\frac{c}{a}=\frac{4}{k}}$

$\implies{(aβ)²=a+²β²+2aβ}$

$\implies{(\frac{4}{k})²=24+2(\frac{4}{k})}$

$\implies{\frac{4²}{k²}=24+2\frac{4}{k}}$

16=24k²+8k

2=3k²+k

0=3k²+k-2

0=3k²+3k-2k-2

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

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

Therefore:-

$\downarrow\huge{\purple{AnsweR}}$

$\huge{k=-1,\frac{2}{3}}$