Nature of the intercepts via the discriminant

Text

Decide whether the parabola $y=x^{2}-4x+7$ meets the $x$-axis.

Solution

Compute the discriminant of $x^{2}-4x+7=0$: $\Delta=(-4)^{2}-4\cdot 1\cdot 7=16-28=-12<0.$ Since $\Delta<0$ and it opens upward, the parabola does not meet the $x$-axis (it lies entirely above it). $\boxed{\,\Delta=-12<0:\ \text{no intersection}\,}$