Question

if Alfa and beta are the zeroes of the quadratic polynomial p(X)=kx²+4x+4 such that alfa²+beta²=24, find the value of k.​

Answer 1

The values of k are 2/3 and -1

Explanation:

Product of zeros = αβ= constant / coefficient of x^2 = 4/k

Sum of zeros =α+β = - coefficient of x / coefficient of x^2= -4/k

Given

`(\alpha)^2 + (\beta)^2 = 24`

`(\alpha)^2 + (\beta)^2` can be written as
`(\alpha)^2 + 2(\alpha)(\beta) + (\beta)^2` if we add
`\pm 2 (\alpha)(\beta)` in the above equation.

`(\alpha)^2 + 2(\alpha)(\beta) + (\beta)^2 -2(\alpha)(\beta)`

`(\alpha + \beta)^2 -2(\alpha)(\beta)`

Putting values of αβ and α+β

`(\frac {-4}{k})^2 -2( \frac {4}{k}) = 24\\\frac {16}{k^2} - \frac {8}{k} = 24\\Multiplying\,\, the \,\, equation\,\, with\,\, 8K^2\\ 2 - k= 3K^2\\3k^2-2+k=0\\or\\3k^2+k-2=0\\3k^2+3k-2k-2=0\\3k(k+1)-2(k+1)=0\\(3k-2)(k+1)=0\\3k-2=0 \,\,and\,\, k+1 =0\\k= 2/3 \,\,and\,\, k=-1`

The values of k are 2/3 and -1