Question
If the difference between a number and its reciprocal is $\frac{24}{5}$, find the number
✓
Answer
Let the number be " $x$ " and its reciprocal is $\frac{1}{x}$
By the given condition
$
\begin{aligned}
& x-\frac{1}{x}=\frac{24}{5} \\
& \frac{x^2-1}{x}=\frac{24}{5} \\
& 5 x^2-5=24 x \\
& 5 x^2-24 x-5=0
\end{aligned}
$
$
\begin{aligned}
& 5 x^2-25 x+x-5=0 \\
& \Rightarrow 5 x(x-5)+1(x-5)=0 \\
& (x-5)(5 x-1)=0 \\
& \Rightarrow x-5=0 \text { or } 5 x+1=0 \\
& x=5 \text { or } 5 x=-1 \\
& \Rightarrow x=\frac{-1}{5}
\end{aligned}
$
The number is 5 or $\frac{-1}{5}$
By the given condition
$
\begin{aligned}
& x-\frac{1}{x}=\frac{24}{5} \\
& \frac{x^2-1}{x}=\frac{24}{5} \\
& 5 x^2-5=24 x \\
& 5 x^2-24 x-5=0
\end{aligned}
$
$
\begin{aligned}
& 5 x^2-25 x+x-5=0 \\
& \Rightarrow 5 x(x-5)+1(x-5)=0 \\
& (x-5)(5 x-1)=0 \\
& \Rightarrow x-5=0 \text { or } 5 x+1=0 \\
& x=5 \text { or } 5 x=-1 \\
& \Rightarrow x=\frac{-1}{5}
\end{aligned}
$
The number is 5 or $\frac{-1}{5}$
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