If $\frac{2 \sqrt{7} + 3 \sqrt{5}}{\sqrt{7} + \sqrt{5}} = P \sqrt{35} + Q$ , then what Is The value of $2 P + Q$ .

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Solution

Step $1$ : Simplify the L.H.S.
Given: $\frac{2 \sqrt{7} + 3 \sqrt{5}}{\sqrt{7} + \sqrt{5}} = P \sqrt{35} + Q$
Take L.H.S and rationalize the denominator. ,
$\frac{2 \sqrt{7} + 3 \sqrt{5}}{\sqrt{7} + \sqrt{5}} = \frac{2 \sqrt{7} + 3 \sqrt{5}}{\sqrt{7} + \sqrt{5}} \times \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} - \sqrt{5}} = \frac{2 (7) - 2 \sqrt{35} + 3 \sqrt{35} - 3 (5)}{7 - 5} = \frac{14 - 15 + \sqrt{35}}{2} = \frac{1}{2} \sqrt{35} - \frac{1}{2}$
Step $2$ : Equate the L.H.S. with the R.H.S.
$\frac{1}{2} \sqrt{35} - \frac{1}{2} = P \sqrt{35} + Q$
Thus, $P = \frac{1}{2}$ and $Q = - \frac{1}{2}$
Step $3$ : Compute the required value.
$2 P + Q = 2 (\frac{1}{2}) - \frac{1}{2} \Rightarrow 2 P + Q = 1 - \frac{1}{2} \Rightarrow 2 P + Q = \frac{1}{2}$
Hence, the value of $2 P + Q$ is $\frac{1}{2}$ .

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