Question

If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f\left(x\right)=x^2-px+q$, then find the values of
(i) ${\alpha }^2+{\beta }^2$
(ii) $\frac{1}{\alpha }+\frac{1}{\beta }$

Answer 1

Consider the given polynomial

$f\left(x\right)=x^2-px+q$

$\alpha$ and $\beta$are the zeros of given polynomial

We know that,

$Sumofzeros$=$-\frac{cofficentofx}{cofficentof{x}^2}$

$\alpha +\beta =\frac{-(-p)}{1}$

$\alpha +\beta =p$

$Productofzeros=\frac{constant}{cofficentof{x}^2}$

$\alpha \times \beta =\frac{q}{1}$

$\alpha \times \beta =q$

Then,

Part (1):-

${\alpha }^2+{\beta }^2={(\alpha +\beta )}^2-2\alpha \beta$

${\alpha }^2+{\beta }^2=p^2-2q$

Part (2):-

$\frac{1}{\alpha }+\frac{1}{\beta }=\frac{\alpha +\beta }{\alpha \beta }$

$=\frac{p}{q}$

Hence, this is the answer.