Question

15. If x = 9 - 4√5, find the value of x² +1/x²​

Answer 1

Answer: x² +1/x²​=322

Step-by-step explanation:

Answer 2

Answer:

x = 9 - 4 \sqrt{5}x=9−45

To find :

x {}^{2} - \frac{1}{x {}^{2} }x2−x21

Solution :

\begin{gathered}x = 9 - 4 \sqrt{5} \\ \\ \frac{1}{x} = \frac{1}{9 - 4 \sqrt{5} } \times \frac{9 + 4 \sqrt{5} }{9 + 4 \sqrt{5} } \\ \\ \frac{1}{x} = \frac{9 + 4 \sqrt{5} }{(9) {}^{2} - (4 \sqrt{5}) {}^{2} } \\ \\ \frac{1}{x} = \frac{9 + 4 \sqrt{5} }{81 - 80} \\ \\ \frac{1}{x} = 9 + 4 \sqrt{5} \end{gathered}x=9−45x1=9−451×9+459+45x1=(9)2−(45)29+45x1=81−809+45x1=9+45

Now,

\begin{gathered}x {}^{2} = (9 - 4 \sqrt{5} ) {}^{2} \\ \\ x {}^{2} = (9) {}^{2} + (4 \sqrt{5}) {}^{2} - 2 \times 9 \times 4 \sqrt{5} \\ \\ x {}^{2} = 81 + 80 - 72 \sqrt{5} \\ \\ x {}^{2} =16 1 - 72 \sqrt{5} \end{gathered}x2=(9−45)2x2=(9)2+(45)2−2×9×45x2=81+80−725x2=161−725

And,

\begin{gathered}( \frac{1}{x} ) {}^{2} = (9 + 4 \sqrt{5} ) {}^{2} \\ \\ \frac{1}{x {}^{2} } =( 9 ){}^{2} + (4 \sqrt{5} ) {}^{2} + 2 \times 2 \times 4 \sqrt{5} \\ \\ \frac{1}{x {}^{2} } = 81 + 80 + 72 \sqrt{5} \\ \\ \frac{1}{x {}^{2} } = 161 + 72 \sqrt{5} \end{gathered}(x1)2=(9+45)2x21=(9)2+(45)2+2×2×45x21=81+80+725x21=161+725

Finally,

\begin{gathered}x {}^{2} - \frac{1}{x {}^{2} } \\ \\ \implies 161 - 72 \sqrt{5} - (161 + 72 \sqrt{5} ) \\ \\ \implies \cancel{161} - 72 \sqrt{5} - \cancel{161} \: - 72 \sqrt{5} \\ \\ \implies - 144 \sqrt{5} \end{gathered}x2−x21⟹161−725−(161+725)⟹161−