Question

if x =√3+√2/√3-√2, find the value of x²​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{x^{2}=49+20\sqrt{6}}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given :}} \\ \tt: \implies x =\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}} \\ \\ \red{\underline \bold{To \: Find :}} \\ \tt: \implies x^{2} =?$

• According to given question :

$\bold{As \: we \: know \: that} \\ \tt: \implies {x}^{2} \\ \\ \tt: \implies (\frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} } ) ^{2} \\ \\ \tt: \implies (\frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} } \times \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} + \sqrt{2} } )^{2} \\ \\ \tt: \implies (\frac{ { (\sqrt{3} + \sqrt{2} )}^{2} }{ { (\sqrt{3}) }^{2} - {( \sqrt{2})}^{2} } )^{2} \\ \\ \tt: \implies (\frac{ {( \sqrt{3} )}^{2} + {( \sqrt{2} )}^{2} + 2 \times \sqrt{3} \times \sqrt{2} }{3 - 2} ) ^{2} \\ \\ \tt: \implies ( \frac{3 + 2 + 2 \sqrt{6} }{1})^{2} \\ \\ \tt: \implies (5 + 2 \sqrt{6} )^{2} \\ \\ \tt: \implies {5}^{2} + {(2 \sqrt{6} )}^{2} + 2 \times 5 \times 2 \sqrt{6} \\ \\ \tt: \implies 25 + 24 + 20 \sqrt{6} \\ \\ \green{\tt: \implies 49 + 20 \sqrt{6} } \\ \\ \green{\tt \therefore {x}^{2} = 49 + 20 \sqrt{6} }$

Answer 2

ANSWER:

Given

• x = √3 + √2/√3 - √2

Squaring on both sides

→ x² = (√3 + √2)²/(√3 - √2)²

Simplifying using

♦ (a + b)² = a² + 2ab + b²

♦ (a - b)² = a² - 2ab + b²

→ x² = (√3)² + 2(√3)(√2) + (√2)²/(√3)² - 2(√3)(√2) + (√2)²

→ x² = 3 + 2√6 + 2/3 - 2√6 + 2

→ x² = 5 + 2√6/5 - 2√6

Simplifying by rationalising denominator

→ x² = (5 + 2√6)(5 + 2√6)/(5 - 2√6)(5 + 2√6)

Using

♦ (a + b)(a - b) = a² - b²

→ x² = 5² + 2(5)(2√6) + (2√6)²/5² - (2√6)²

→ x² = 25 + 20√6 + 24/25 - 24

→ x² = 49 + 20√6

Hence, x² = 49 + 20√6