Question

if x=2-√3, then what is x²+1/x²

Answer 1

x = 2-√3
fill the value in
x²+1/x²
use formula (a-b)²= a²+b²-2ab
=x²+1 / x²

=(2-√3)² + 1/(2-√3)²

=(2)²+(√3)²-2(2)(√3) +1 / (2)²+(√3)²-2(2)(√3)

= 4+3-4√3+1 / 4+3+-4√3

= 8-4√3 / 7-4√3

= 4(2-√3) / 7-4√3(common)

Answer 2

Given x=2−3–√x=2−3

Find x2−x−2x2−x−2

x2−x−2=x2−1x2x2−x−2=x2−1x2

x2=(2−3–√)2=4−43–√+3=7−43–√x2=(2−3)2=4−43+3=7−43

x2−x−2=7−43–√−17−43–√x2−x−2=7−43−17−43

Use the difference of squares trick to rationalize the denominator.

x2−x−2=7−43–√−7+43–√(7−43–√)(7+43–√)x2−x−2=7−43−7+43(7−43)(7+43)

=7−43–√−7+43–√(49−16⋅3)=7−43−7+43(49−16⋅3)

=7−43–√−7+43–√(49−48)=7−43−7+43(49−48)

=7−43–√−(7+43–√)=7−43−(7+43)

=7−43–√−7−43–√=7−43−7−43

=−83–√