For what values does $kx^2 - 6x - 4 = 0$ cut the x-axis?

Question

A quadratic equation problem.

For what values does $kx^2 - 6x - 4 = 0$ cut the x-axis?

The preceding part of this question dealt with the discriminant.

Answer 1

Assumingly you mean "For what values of $k$ does $f(x)=kx^2-6x-4$ cut the x-axis?".

A continuous function cuts the x-axis exactly at the points $x=x_i$ where $f(x_i)=0$ (and where $f(x)$ doesn't "touch" the x-axis).

The question is equivalent to "For what values of $k$ does the equation $kx^2-6x-4=0$ have a solution $x=x_i$ at which point $f(x)=kx^2-6x-4$ doesn't "touch" the x-axis?".

And a quadratic equation has a "non-touching" solution if and only if $\Delta=(-6)^2-4(k)(-4)> 0$ (the "touching" solution is if and only if $\Delta=0$).

But remember that the equation is only quadratic when $k\neq 0$. If $k=0$, the equation becomes $-6x-4=0$. Does this one have a solution?