Question

Let f(x) be a quadratic polynomial satisfying f(2)+f(4)=0. If unity is one root of f(x)=0 then find other root

Answer 1

Given:

f(x) is a quadratic polynomial

f(2)+f(4) = 0

And x=1 is a root of f(x)

To find:

We have to find the other root of quadratic equation

Solution:

In general a quadratic polynomial f(x) is given by $f(x)=ax^{2} +bx+c$ where  a,b and c are constants  and roots of polynomial be $\alpha$ and $\beta$

We have the relation between roots as $\alpha +\beta =-\frac{b}{a}$   and   $\alpha \beta =\frac{c}{a}$

Let one root be $\alpha =1$  and other root be  $\beta$

Now, f(x) can be written as  $f(x)= x^{2} -(-\frac{b}{a} )x+\frac{c}{a}$

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

                                             $f(x)=x^{2} -(1+\beta )x+ \beta$

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

It is given that f(2)+f(4) = 0

⇒$[2^{2}-2-2\beta +\beta ]+[4^{2}-4-4\beta +\beta ]=0$

⇒$[4-2-\beta ]+[16-4-3\beta ]=0$

⇒$2-\beta +12-3\beta=0$

⇒$24-4\beta =0$

⇒$4\beta =24$

⇒$\beta =6$

∴The other root of f(x) is 6.