Question

Determine rational number p and q if:7+√5/7-√5-7-√5/7+√5=p-7√5q​

Answer 1

*Answer:-

The value of

• p = 0

and

• q = -2

Answer 2

Answer:

$\displaystyle{\boxed{\red{\sf\:p\:=\:0}}\sf\:\quad\:\&\:\quad\:\boxed{\red{\sf\:q\:=\:\dfrac{1}{11}}}}$

Step-by-step-explanation:

We have given that,

$\displaystyle{\sf\:\dfrac{7\:+\:\sqrt{5}}{7\:-\:\sqrt{5}}\:-\:\dfrac{7\:-\:\sqrt{5}}{7\:+\:\sqrt{5}}\:=\:p\:-\:7\:\sqrt{5}\:q}$

We have to determine p and q.

Now,

$\displaystyle{\sf\:\dfrac{7\:+\:\sqrt{5}}{7\:-\:\sqrt{5}}\:-\:\dfrac{7\:-\:\sqrt{5}}{7\:+\:\sqrt{5}}\:=\:p\:-\:7\:\sqrt{5}\:q}$

$\displaystyle{\implies\sf\:\dfrac{(\:7\:+\:\sqrt{5}\:)\:\times\:(\:7\:+\:\sqrt{5}\:)\:-\:(\:7\:-\:\sqrt{5}\:)\:\times\:(\:7\:-\:\sqrt{5}\:)}{(\:7\:-\:\sqrt{5}\:)\:(\:7\:+\:\sqrt{5}\:)}\:=\:p\:-\:7\:\sqrt{5}\:q}$

$\displaystyle{\implies\sf\:\dfrac{(\:7\:+\:\sqrt{5}\:)^2\:-\:(\:7\:-\:\sqrt{5}\:)^2}{(\:7\:)^2\:-\:(\:\sqrt{5}\:)^2}\:=\:p\:-\:7\:\sqrt{5}\:q}$

$\displaystyle{\implies\sf\:\dfrac{(\:7\:+\:\cancel{\sqrt{5}}\:+\:7\:-\:\cancel{\sqrt{5}}\:)\:(\:\cancel{7}\:+\:\sqrt{5}\:-\:\cancel{7}\:+\:\sqrt{5}\:)}{49\:-\:5}\:=\:p\:-\:7\:\sqrt{5}\:q}$

$\displaystyle{\implies\sf\:\dfrac{14\:\times\:\cancel{2}\:\sqrt{5}}{\cancel{44}}\:=\:p\:-\:7\:\sqrt{5}\:q}$

$\displaystyle{\implies\sf\:\dfrac{\cancel{14}\:\sqrt{5}}{\cancel{22}}\:=\:p\:-\:7\:\sqrt{5}\:q}$

$\displaystyle{\implies\sf\:\dfrac{7\:\sqrt{5}}{11}\:=\:p\:-\:7\:\sqrt{5}\:q}$

$\displaystyle{\implies\sf\:0\:-\:\dfrac{1}{11}\:7\:\sqrt{5}\:=\:p\:-\:7\:\sqrt{5}\:q}$

$\displaystyle{\implies\underline{\boxed{\red{\sf\:p\:=\:0}}}\sf\:\quad\:\&\:\quad\:\underline{\boxed{\red{\sf\:q\:=\:\dfrac{1}{11}}}}\:\sf\:\quad\:\dots\:[\:By\:compairing\:]}$