Question

If $\alpha $ and $\beta $ are the zeros of the quadratic polynomial such that $\alpha + \beta = 24$ and $\alpha - \beta = 8$. Find a quadratic polynomial having a $\alpha $ and $\beta $ as its zeros.

Answer 1

Hint-Try to put both the quadratic polynomial equations given in the question. By comparing them and simplifying it, we will get the value which you can also cross check by putting them again in the given quadratic polynomial equation.

Complete step-by-step answer:
According to the question, we have given two equations $\left( i \right)$ and $\left( {ii} \right)$,
$\alpha + \beta = 24....eq\left( i \right)$
$\alpha - \beta = 8....eq\left( {ii} \right)$
By comparing, both the equation, we will get the values of $\alpha $ and $\beta $ respectively,
$ \Rightarrow 2\alpha = 32$
$ \Rightarrow \alpha = 16$
By putting the value of $\alpha $ in eq $\left( {ii} \right)$, we will get
$ \Rightarrow \alpha - \beta = 16$
$ \Rightarrow 16 - \beta = 8$
$ \Rightarrow \beta = 8$
So, $\alpha = 16$ and $\beta = 8$
Also, we will calculate the product of $\alpha $and $\beta $, we will get
$ \Rightarrow \alpha \beta = 128$
As, we have all the values now we will put it in the quadratic polynomial equation,
$\left[ {{x^2} - \left( {\alpha + \beta } \right)x + \alpha \beta } \right]$
$ \Rightarrow \left[ {{x^2} - \left( {24} \right)x + 128} \right]$

Note-These types of questions are very easy as long as we have prior knowledge of the concept of quadratic polynomial equations. In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function with one or more variables in which the highest-degree term is of the second degree.