Roots and the discriminant.

Step 1 of 4

Where the graph meets the axis

The roots of $y = ax^2 + bx + c$⁠ are the values of $x$⁠ where $y = 0$⁠: the points where the parabola crosses the $x$⁠-axis. Look inside is open and marks them in orange. The quadratic formula finds them:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$x =$$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$⁠

The expression under the square root is the discriminant, $\Delta = b^2 - 4ac$⁠. The Scene shows it next to the roots. Move c and keep an eye on $\Delta$⁠ and on the number of roots.

Step 2 of 4

Predict before you look

The coefficients are locked. Raising $c$⁠ lifts the whole parabola. Will it still cross the axis?

Step 3 of 4

Work out the discriminant

This time Look inside is hidden, so the Scene cannot give the answer away. Work it out yourself.

Step 4 of 4

When the discriminant is negative

There are three cases, depending on the sign of $\Delta$⁠:

• $\Delta > 0$⁠: two distinct roots; the parabola crosses the axis twice.
• $\Delta = 0$⁠: one (double) root; the vertex touches the axis.
• $\Delta < 0$⁠: no real roots; the parabola lies entirely above the axis or entirely below it.