How to take "a" as positive and "b" as negative root of quadratic equation in this question?

Question

Hi, I am not getting how to take "a" as negative root and "b" as positive root, while I am only having X in my quadratic equation. What is the connection of a and b with this quadratic equation in this question?

My approach to this question: I solved it and taken out two roots as -6 and +1, now I am getting confused as I am not having any a and b in a quadratic equation.

Answer 1

It factors as $(x-1)(x+6)$ so the positive root is $1$ and the negative root is $-6.$ Plug these into your expression(s).

Answer 2

Hint: The expression reduces to $$ \frac{a}{b}+\frac{b}{a} = \frac{a^2+b^2}{ab} = \frac{(a+b)^2-2ab}{ab} $$ and you know $a+b$ and $ab$ from the coefficients of $x^2+5x-6=0$.

The talk about positive and negative roots is a red herring because the original expression is symmetric in $a$ and $b$, even though it does not look symmetric.