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​

Given:

[tex]x^2+kx-98=0[/tex]

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 [tex]\alpha[/tex] and [tex]\beta[/tex].

According to the question,

[tex]\beta=2(-\alpha)[/tex]        (Additive inverse of [tex]\alpha[/tex] is [tex]-\alpha[/tex])

[tex]\beta=-2\alpha[/tex]

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

[tex]\alpha+\beta=-\dfrac{b}{a}[/tex]

[tex]\alpha\beta=\dfrac{c}{a}[/tex]

We have,

[tex]x^2+kx-98=0[/tex]

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

[tex]\alpha+\beta=-\dfrac{k}{1}[/tex]

[tex]\alpha-2\alpha=-k[/tex]           [tex][\because \beta=-2\alpha][/tex]

[tex]-\alpha=-k[/tex]

[tex]\alpha=k[/tex]           ...(i)

Now,

[tex]\alpha\beta=\dfrac{c}{a}[/tex]

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

[tex]-2\alpha^2=-98[/tex]

Divide both sides by -2.

[tex]\alpha^2=49[/tex]

Taking square root on both sides.

[tex]\alpha=\pm \sqrt{49}[/tex]

[tex]\alpha=\pm 7[/tex]

Using (i), we get

[tex]k=\pm 7[/tex]

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

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Mount Everest Great Himalayas of southern Asia 8,849.87 meters
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Mount Olympus Mytikas, Greece 2,919.07 meters
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Part B:

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