Hi.

I guess you know how to sketch y = x-1.

What about the quadratic function k(x^2) ? I guess you know it has a maximum or minimum in x = 0.

if k < 0 the parabola opens downward. So the curve kx^2 has a maximum in x=0.

If you substitute x = 0 in y = x-1 it is 0-1 = -1, so the line is below the quadratic function and there got to be two solutions, because y = x-1 is a linear function.

Otherwise you can show it directly:

Using the quadratic equation leads to

Assume k < 0. There are two solutions, if

Of course this is > 0, because let k:= -1

then

and

If you substitute -a = k (if k=-1 the a is equal to -(-1) = +1), you get

This could be kinda confusing...

Yours

Rapha