Question

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

the additive inverse of the other.
«+
C​

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=-(b)/(a)`

`\alpha\beta=(c)/(a)`

We have,

`x^2+kx-98=0`

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

`\alpha+\beta=-(k)/(1)`

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

`-\alpha=-k`

`\alpha=k` ...(i)

Now,

`\alpha\beta=(c)/(a)`

`\alpha(-2\alpha)=(-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 √(49)`

`\alpha=\pm 7`

Using (i), we get

`k=\pm 7`

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