Question

Find the value of k , if one of the two roots of the equation : x2 + k X – 98 = 0 is double

Answer 1

Given:

$x^2+kx-98=0$

One of the two roots of the given equation is double  the additive inverse of the other.

To find:

The value of k.

Solution:

Let two roots of the given equation are $\alpha$ and $\beta$.

According to the question,

$\beta=2(-\alpha)$        (Additive inverse of $\alpha$ is $-\alpha$)

$\beta=-2\alpha$

If $\alpha$ and $\beta$ are roots of the quadratic equation $ax^2+bx+c=0,$ then

$\alpha+\beta=-\dfrac{b}{a}$

$\alpha\beta=\dfrac{c}{a}$

We have,

$x^2+kx-98=0$

Here, a =1, b=k and c=-98.

$\alpha+\beta=-\dfrac{k}{1}$

$\alpha-2\alpha=-k$           $[\because \beta=-2\alpha]$

$-\alpha=-k$

$\alpha=k$           ...(i)

Now,

$\alpha\beta=\dfrac{c}{a}$

$\alpha(-2\alpha)=\dfrac{-98}{1}$          $[\because \beta=-2\alpha]$

$-2\alpha^2=-98$

Divide both sides by -2.

$\alpha^2=49$

Taking square root on both sides.

$\alpha=\pm \sqrt{49}$

$\alpha=\pm 7$

Using (i), we get

$k=\pm 7$

Therefore, the value of k is either -1 or 1.