Question

If $p$ and $q$ are rational numbers, then find $p$ and $q$ given that $\frac{4+\sqrt{14}}{4-\sqrt{14}}=p+q\sqrt{14}$

• A. $p=15,q=4$
• B. $p=-15,q=4$
• C. $p=15,q=-4$
• D. $p=-15,q=-4$

Answer 1

The correct option is A $p=15,q=4$

$\frac{4+\sqrt{14}}{4-\sqrt{14}}=p+q\sqrt{14}$

$\frac{4+\sqrt{14}}{4-\sqrt{14}}\times \frac{4+\sqrt{14}}{4+\sqrt{14}}=p+q\sqrt{14}$

$\frac{(4+\sqrt{14}{)}^2}{(4{)}^2-(\sqrt{14}{)}^2}=p+q\sqrt{14}$

$\frac{(4{)}^2+2(4\left)\right(\sqrt{14})+(\sqrt{14}{)}^2}{16-14}=p+q\sqrt{14}$

$\frac{16+8\sqrt{14}+14}{2}=p+q\sqrt{14}$

$\frac{30+8\sqrt{14}}{2}=p+q\sqrt{14}$

$15+4\sqrt{14}=p+q\sqrt{14}$

⇒ $p=15,q=4$