Question

if x = 4 + √ 15 find the value of x ² + 1/ x²​

Answer 1

Given:

★ x = 4+√15.

To Find:

★$\sf{\dfrac{1}{x^{2}}+x^{2}}$

Answer:

We have,

$\sf{\implies x =4+\sqrt{15}}$

$\sf{\implies \dfrac{1}{x}=\dfrac{1}{4}+\sqrt{15}}$

$\sf{\implies \dfrac{1}{x}=\dfrac{4-\sqrt{15}}{(4+\sqrt{15})(4-\sqrt{15})}}$

$\sf{\implies \dfrac{1}{x}=\dfrac{4-\sqrt{15}}{16-15}}$

using

• (a+b)(a-b) = a²-b²

$\sf{\red{\leadsto\dfrac{1}{x}=4-\sqrt{15}}}$

$\rule{200}1$

So,

x²=(4+√15)²=16+15+8√15 = 31+8√15

1/x²= (4-√15)²=16+15-8√15 =31-8√15.

using

• (a+b)²=a²+b²+2ab
• (a-b)²=a²+b²-2ab.

$\rule{200}1$

.°. x²+1/x²

=31+8√1531-8√15

=62.