Find the value of p and q where p and q are rational numbers.

If {[7 + 3√5] / [3 + √5]} – {[7 – 3√5] / [3 – √5]} = p + q√5, find the value of p and q where p and q are rational numbers.

Solution:

Let us consider LHS
Since there are two terms, let us solve individually
{[7 + 3√5] / [3 + √5]}
Rationalize the denominator,
[7 + 3√5] / [3 + √5] = [(7 + 3√5) (3 – √5)] / [(3 + √5) (3 – √5)]
= [7(3 – √5) + 3√5(3 – √5)] / [32 – (√5)2]
= [21 – 7√5 + 9√5 – 3.5] / [9 – 5]
= [21 + 2√5 – 15] / [4]
= [6 + 2√5] / 4
= 2[3 + √5]/4
= [3 + √5] /2
Now,
{[7 – 3√5] / [3 – √5]}
Rationalize the denominator,
[7 – 3√5] / [3 – √5] = [(7 – 3√5) (3 + √5)] / [(3 – √5) (3 + √5)]
= [7(3 + √5) – 3√5(3 + √5)] / [32 – (√5)2]
= [21 + 7√5 – 9√5 – 3.5] / [9 – 5]
= [21 – 2√5 – 15] / 4
= [6 – 2√5]/4
= 2[3 – √5]/4
= [3 – √5]/2
So, according to the question
{[7 + 3√5] / [3 + √5]} – {[7 – 3√5] / [3 – √5]}
By substituting the obtained values,
= {[3 + √5] /2} – {[3 – √5] /2}
= [3 + √5 – 3 + √5]/2
= [2√5]/2
= √5
So when comparing with RHS
√5 = p + q√5
Hence, value of p = 0 and q = 1

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