Roots of a Quadratic on a Graph

Today we find where a parabola crosses the x-axis—those are the roots. A quadratic function is y equals a x squared plus b x plus c.

The roots are the x-values where y equals zero—where the curve touches the x-axis. Let's use a concrete example: y equals x squared minus five x plus six.

• Roots: y = 0, curve meets x-axis

I'll plot this parabola so you see where it crosses the x-axis. See? The parabola crosses the x-axis at x equals two and x equals three. Those two points are the roots of this quadratic.

• Roots are x = 2 and x = 3

We can verify: plug in x equals two. Two squared minus ten plus six equals zero. Check.

A parabola can have two roots, one root, or no real roots at all. Two roots means the parabola crosses the x-axis twice—like our example. One root means the vertex just touches the x-axis at the top or bottom.

• Two roots, one root, or none
• Two roots: parabola crosses twice
• One root: vertex touches x-axis

No real roots means the parabola never touches the x-axis—it stays above or below. Quick check: for y equals x squared minus five x plus six, how many roots do you see? The roots are where the parabola meets the x-axis—find them by graphing or solving.

• No roots: parabola stays above/below
• Roots found graphically or algebraically