Question

if x=5+√7 find the value of x squade+1/xsquare​

Answer 1

Given :-

x = 5 + √7

➡ x² = (5 + √7)²

using identity (a + b)² = a² + 2ab + b²

= (5)² + 2(5)(√7) + (√7)²

= 25 + 10√7 + 7

= 32 + 10√7

➡ 1/x² = 1/(32 + 10√7)

by rationalizing, we get

$\tt = \frac{1}{32 + 10 \sqrt{7} } \times \frac{32 - 10 \sqrt{7} }{32 - 10 \sqrt{7} } \\ \\ \tt = \frac{32 - 10 \sqrt{7} }{(32 + 10 \sqrt{7})(32 - 10 \sqrt{7} ) } \\ \\ \tt = \frac{32 - 10 \sqrt{7} }{( {32)}^{2} - ( {10 \sqrt{7} })^{2} } \\ \\ \tt = \frac{32 - 10 \sqrt{7} }{1024 - 700} \\ \\ \tt = \frac{32 - 10 \sqrt{7} }{324} \\ \\ \tt = \frac{16 + 5 \sqrt{7} }{162}$

hence, value of x² + 1/x² :-

= 32 + 10√7 + (16 + 5√7)/162

= 162(30 + 10√7)/162 + (16 + 5√7)/162

= (4860 + 1620√7)/162 + (16 + 5√7)/162

= (4876 + 1625√7)/162