Question

if alpha beta are zeros of p(x) =kx square + 4 x + 4 such that Alpha square plus beta square is equals to 4 find the value of k​

Answer 1

Answer:

Step-by-step explanation:

α and β are the zeros of the given polynomial Kx² + 4x + 4 = 0

so, product of zeros = αβ = constant/coefficient of x² = 4/K

sum of zeros = α + β = -coefficient of x/Coefficient of x² = -4/k

Now, α² + β² = 24

⇒(α + β)² - 2αβ = 24

⇒(-4/k)² - 2(4/k) = 24

⇒16/K² - 8/k = 24

⇒ 2 - k = 3k²

⇒3k² + k -2 = 0

⇒ 3k² + 3k - 2k - 2 = 0

⇒3k(k + 1) - 2(k +1) = 0

⇒(3k -2)(k + 1) = 0

Hence, k = 2/3 and -1

Answer 2

Answer:

p(x) = kx² + 4x + 4

if α, β are the roots of the above equation then

α + β =  - 4/k and αβ = 4/k

α² + β² = 4 (given)

(α + β)² = α² + β² + 2αβ

(-4/k)² = 4 + 2*4/k

16/k² = 4 + 8/k

16/k² = (4k + 8)/k

16k = 4k³ + 8k²

16k - 4k³ - 8k² = 0

-4k(k² + 2k - 4) = 0

either k = 0 but k≠0 because it is a coefficient of x²

or k² + 2k - 4 = 0 where  a = 1, b = 2, c = -4

   k = -2 ± √(4 + 16) / 8

   k = (-2 ± √20)/8

   k = (-2±2√5)/8

   k = (-1±√5)/4