Question

if x=5-root24 then find 10{x^2+1/x^2}-97{x+1/x}-10

Answer 1

The key thing is to use the identity
$x^2+\frac{1}{x^2}=\left(x+\frac{1}{x}\right)^2-2$

In light of that, start by working the value of $x+\frac{1}{x}$ : 
$x=5-\sqrt{24}$
$\frac{1}{x}=\frac{1}{5-\sqrt{24}}=\frac{1}{5-\sqrt{24}}\times\frac{5+\sqrt{24}}{5+\sqrt{24}}=\frac{5+\sqrt{24}}{5^2-\sqrt{24}^2}=\frac{5+\sqrt{24}}{25-24}=5+\sqrt{24}$

Add them and get
$x+\frac{1}{x}=5-\sqrt{24}+5+\sqrt{24}=10$

Use the earlier identity to find the value of $x^2+\frac{1}{x^2}$ :
$x^2+\frac{1}{x^2}=\left(x+\frac{1}{x}\right)^2-2=10^2-2=98$

Plug these values in the given expression and simplify :
$10\left(x^2+\frac{1}{x^2}\right)-97\left(x+\frac{1}{x}\right)-10\\=10\left(98\right)-97\left(10\right)-10\\=\boxed{0}$

Answer 2

If
x=5-√24
1/x=1/5-√24
on rationalize
1/x=1        × 5+√24    =5+√24  =5+√24
     5-√24      5+√24    25-25

on looking identity (x+1/x)²=x²+1/x²+2
x²+1/x²=(x+1/x)²-2
on putting value
(5-√24 +5+√24)² -2
100-2=98
According to question
10{x²+1/x²} -97{x+1/x)-10
10(98) -97(10)-10
980-970-10⇒980-980=0
Answer is o