Question

If alfa and bita are the zeroes of the quadratic polynomial f(x)=kx^2+4x+4 such that alfa^2 +bita^2=24 find the value of k​

Answer 1

Correct Question : If α and β are zeroes of the quadratic Polynomial f(x) = kx² + 4x + 4 such that α² + β² = 24. Find the value of k.

SOLUTION :

Given,

Quadratic Equation :

f(x) = kx² + 4x + 4

α² + β² = 24

We know that,

Sum of zeroes using coefficients :

α + β = -b/a

α + β = -4/k

Product of zeroes using coefficients :

αβ = c/a

αβ = 4/k

α² + β² = 24

α² + β² = (α + β)² - 2αβ

(α + β)² - 2αβ = 24

(-4/k)² - 2(4/k) = 24

16/k² - 8/k = 24

8 [ 2/k² - 1/k ] = 24

2/k² - 1/k = 24/8

2/k² - 1/k = 3

2 - k = 3k²

3k² + k - 2 = 0

3k² + 3k - 2k - 2 = 0

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

k + 1 = 0 ; 3k - 2 = 0

k = -1 ; 3k = 2

k = -1 ; k = 2/3

Therefore, k = -1 or 2/3.

Answer 2

f(x)= kx^2 + 4x +4

alpha + beta = -4/k

alpha × beta = 4/k

alpha^2 + beta^2= ( alpha+ beta)^2 - 2 alpha × beta

24 = 16/k^2 - 2( 4/k)

24 = 16/k^2 - 8/k

3k^2 + k - 2 = 0

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

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

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

k= 2/3,-1