Question

If difference between the roots of the equation x^2-kx +8 =0 is 4 then the value of K is

Answer 1

Answer:-

The value of k is $\sf{4\sqrt3}$.

Given:

• The given quadratic equation is $\sf{x^{2}-kx+8=0}$

• Difference between the roots of equation is 4.

To find:

• The value of k

Solution:-

The given quadratic equation is $\sf{x^{2}-kx+8=0}$

Let the roots be $\sf{\alpha \ and \ \beta}$

$\sf{\therefore{\alpha-\beta=4...(1)}}$

Sum of roots=$\sf{\frac{-b}{a}}$

$\sf{\therefore{\alpha+\beta=k...(2)}}$

Product of roots=$\sf{\frac{c}{a}}$

$\sf{\therefore{\alpha\beta=8...(3)}}$

According to the identity

$\sf{(a+b)^{2}=(a-b)^{2}+4ab}$

$\sf{(\alpha+\beta)^{2}=(\alpha-\beta)^{2}+4\alpha\beta}$

From (1), (2) and (3)

$\sf{k^{2}=4^{2}+4(8)}$

$\sf{k^{2}=16+32}$

$\sf{k^{2}=48}$

On taking square root of both sides.

$\sf{k=\sqrt48}$

$\sf{\therefore{k=4\sqrt3}}$

The value of k is $\sf{4\sqrt3}$.

Answer 2

$\purple{\underline{\underline{\pink{\boxed{\boxed{\red{\star{\sf Question:}}}}}}}}$

$\rm{If \ difference \ between \ the \ roots \ of \ the }$

$\rm{equation \ x^2-kx +8 =0 \ is \ 4 \ then\ find}$

$\rm{the \ value \ of \ k }$

_________________________________________

$\purple{\underline{\underline{\orange{\boxed{\boxed{\green{\star{\sf Answer:}}}}}}}}$

$\rm{\red{\boxed{\blue{The \ value \ of \ k \ is \ 4\sqrt3}}}}$

_________________________________________

$\sf\large\underline\pink{Given:}$

$\rm{The \ given \ quadratic \ equation \ is}$

$\rm{x^{2}-kx+8=0}$

$\rm{Difference \ between \ the \ roots \ of \ equation }$

$\rm{ is \ 4.}$

_________________________________________

$\sf\large\underline\blue{To \ Find}$

$\rm{The \ value \ of \ k}$

_________________________________________

$\purple{\underline{\underline{\blue{\boxed{\boxed{\pink{\star{\sf Solution:}}}}}}}}$

$\rm{The \ given \ quadratic \ equation \ is}$

$\rm{x^{2}-kx+8=0}$

$\rm{Difference \ between \ the \ roots \ of \ equation }$

$\rm{ is \ 4.}$

$\rm{Let \ the \ roots \ be \ \alpha \ and \ \beta}$

$\rm{\therefore{\alpha-\beta=4.......(a)}}$

$\rm{We \ know ,}$

$\rm{Sum \ of \ roots \ \frac{-b}{a}}$

$\rm{\therefore{\alpha+\beta=k.......(b)}}$

$\rm{Also,}$

$\rm{Product \ of \ roots \ = \ \frac{c}{a}}$

$\rm{\therefore{\alpha\beta=8.........(c)}}$

$\rm{(a+b)^{2}=(a-b)^{2}+4ab}$

$\rm{(According \ to \ the \ identity )}$

$\rm{(\alpha+\beta)^{2}=(\alpha-\beta)^{2}+4\alpha\beta}$

$\rm{From \ (a), \ (2) \ and \ (3)}$

$\rm\therefore{k^{2}=4^{2}+4(8)}$

$\rm\therefore{k^{2}=16+32}$

$\rm\therefore{k^{2}=48}$

$\rm{On \ taking \ square \ root \ of \ both \ sides. }$

$\rm{k=\sqrt48}$

$\rm{\therefore{k=4\sqrt3}}$

$\rm{\red{\boxed{\blue{The \ value \ of \ k \ is \ 4\sqrt3}}}}$

____________________________________________

$\rm\red{Hope \ it \ Helps \ :))}$