Mathematics

The key insight is that "touching" the $x$-axis means the parabola has exactly one $x$-intercept—the discriminant must equal zero. Since you have two different quadratics with the same leading coefficient and constant term but different $k$ values, you're really finding which values of $k$ make the discriminant $b^2 - 4ac = 0$. This gives you a quadratic equation in $k$ itself, which will have two solutions corresponding to your two different functions.

The key insight is that "touching" the $x$-axis means the parabola has exactly one $x$-intercept—the discriminant must equal zero. Since you have two different quadratics with the same leading coefficient and constant term but different $k$ values, you're really finding which values of $k$ make the discriminant $b^2 - 4ac = 0$. This gives you a quadratic equation in $k$ itself, which will have two solutions corresponding to your two different functions.