Question

Two non-zero real numbers, a and b, satisfy ab=a-b. Find a possible value of
`a/b +b/a -ab`

100 points

Answer 1

2

Explanation:

Given equations:

`\begin{cases}ab=a-b\\\\(a)/(b)+(b)/(a)-ab\end{cases}`

Rewrite the second equation as a rational expression:

`\begin{aligned}\implies (a)/(b)+(b)/(a)-ab & = (a)/(b) \cdot (a)/(a)+(b)/(a) \cdot (b)/(b)-ab\cdot (ab)/(ab) \\\\&=(a^2)/(ab)+(b^2)/(ab)-((ab)^2)/(ab)\\\\&=(a^2+b^2-(ab)^2)/(ab)\\\\\end{aligned}`

Substitute the first equation
`ab=a-b` into the rational expression:

`\begin{aligned}\implies (a^2+b^2-(a-b)^2)/(ab)&=(a^2+b^2-(a^2-2ab+b^2))/(ab)\\\\&=(a^2+b^2-a^2+2ab-b^2)/(ab)\\\\&=(2ab)/(ab)\\\\&=2\end{aligned}`

Therefore, the possible value of the given expression is 2.