Question

if alpha and beta are the zeroes of the quadratic polynomial: p(x)= kx^2+4x+4 such that alpha^2 + beta^2= 24, find the value of K

Answer 1

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α and β are the zero of the Kx² + 4x + 4 , α² + β² = 24 then find k ?

Solution :- 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 + 6k - k - 2 = 0
⇒3k(k + 2) - 1(k +2) = 0
⇒(3k -1)(k +2) = 0
Hence, k = 1/3 and -2

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