Question

If one of the zeros of quadratic polynomial $f\left(x\right)=14x^2-42k^{2}x-9$ is negative of the other, find the value of $k$.

Answer 1

Let $\alpha (\ne 0)$ be one zero of the quadratic polynomial $f\left(x\right)=14x^2-42k^{2}x-9$.

Then $f\left(\alpha \right)=0$

or, $14{\alpha }^2-42k^2\alpha -9=0$.......(1).

Then according to the problem,

$f(-\alpha )=0$

or, $14{\alpha }^2+42k^2\alpha -9=0$......(2).

Now, subtracting (2) from (1) we get,

$84k^2\alpha =0$

or, $k^2=0$ [ Since $\alpha \ne 0$]

or, $k=0$.