If $α$ and $β$ are the zeros of a quadratic polynomial such that $α\ +\ β\ =\ 24$ and $α\ -\ β\ =\ 8$, find a quadratic polynomial having $α$ and $β$ as its zeros.

Given:

$α$  and  $β$  are the zeros of a quadratic polynomial such that  $α\ +\ β\ =\ 24$  and  $α\ -\  β\ =\ 8$.

To do:

We have to find the quadratic polynomial having $α$ and $β$ as its zeros.  

Solution:

$α+β= 24$ ---(1)

$α\ -\  β\ =\ 8$----(2)

Adding equations (1) and (2),

$α+β+α-β=24+8$

$2α=32$

$α=16$

Substituting $α=16$ in equation (1), we get,

$16+β= 24$

$β= 24-16=8$

The quadratic polynomial having roots $α$ and $β$ is $f(x)=k(x^2-(α+β)x+αβ)$, where $k$ is any non-zero real number.

Therefore,

$f(x)=k(x^2-(24)x+(16)(8))$

$f(x)=k(x^2-24x+128)$

The required quadratic polynomial is $f(x)=k(x^2-24x+128)$, where $k$ is any non-zero real number. 

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Updated on: 10-Oct-2022

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