Question

Determine rational numbers a and b.
4+3V5 = a + b 2
4-315​

Answer 1

Correct Question :-

Determine the rational numbers a and b if ,

$\: \frac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5} } = a + b\sqrt{5}$

Solution :-

$\longrightarrow \: \frac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5} } = a + b \sqrt{5}$

Divide and multiply in the LHS with (4 + 3√5) →

$\longrightarrow \frac{(4 + 3 \sqrt{5})(4 + 3 \sqrt{5}) }{(4 - 3 \sqrt{5})(4 + 3 \sqrt{5} ) } = a + b \sqrt{5} \\ \\ \longrightarrow \frac{( {4 + 3 \sqrt{5} )}^{2} }{( {4)}^{2} - ( {3 \sqrt{5}) }^{2} } = a + b \sqrt{5} \\ \\ \longrightarrow \frac{16 + 45 + 2 \times 4 \times 3 \sqrt{5} }{16 - 45} = a + b \sqrt{5} \\ \\ \longrightarrow \frac{61 + 24 \sqrt{5} }{ - 29} = a + b \sqrt{5} \\ \\ \longrightarrow \frac{61}{ - 29} + \frac{24 \sqrt{5} }{ - 29} = a + b \sqrt{5}$

By Comparing both sizes →

$\longrightarrow \: a = - \frac{61}{29} \\ \\\longrightarrow b = - \frac{24 \sqrt{5} }{29}$