Question

$4x^2-x-k$ is a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

$\frac{1}{4},-1$. Find the value of $k$.

Answer 1

Let the quadratic polynomial be $a{x}^2+bx+c$

Then Sum of roots = $-\frac{b}{a}=\frac{1}{4}$

and Product of roots = $\frac{c}{a}=-1$

i.e., $\frac{b}{a}=\frac{-1}{4}$ and $\frac{c}{a}=\frac{-1}{1}$

We select a $=LCM(4,1)=4$

Then $\frac{b}{4}=\frac{-1}{4}$ and $\frac{c}{4}=-1$

$\Rightarrow b=-1$ and $c=-4$

Substituting $a=4,b=-1,c=-4$ in $a{x}^2+bx+c$,

we get the required polynomial $f\left(x\right)=(4x^2-x-4)$

Comparing with given polynomial $4x^2-x-k$, we get $k=4$.