Discriminant

What its sign tells you

Discriminant	Real solutions	The parabola
$\Delta > 0$	two	crosses the $x$-axis twice
$\Delta = 0$	one, repeated	touches the axis at its vertex
$\Delta < 0$	none	never reaches the axis

Why the sign decides it

The quadratic formula ends in $\pm\sqrt{b^2 - 4ac}$, so the discriminant is whatever sits under that root.

• A positive value has two square roots, one positive and one negative, so $\pm$ produces two different answers.
• Zero has one square root, namely zero, so $+0$ and $-0$ give the same answer twice.
• A negative value has no real square root, so the formula produces no real answer.

The three cases are not a rule to memorise. They are the three things a square root can do.

Worked examples

How many solutions does $x^2 + 4x + 4 = 0$ have?

$$\Delta = 16 - 4(1)(4) = 0$$

One repeated solution. The graph touches the $x$-axis without crossing, and indeed $x^2 + 4x + 4 = (x+2)^2$, giving $x = -2$ twice.

How many solutions does $3x^2 + x + 5 = 0$ have?

$$\Delta = 1 - 4(3)(5) = 1 - 60 = -59$$

Negative, so no real solutions. Worth computing first — it saves you from taking the square root of a negative partway through.

A perfect square is a hint

If $\Delta$ is a perfect square such as $1$, $4$ or $25$, the solutions are rational and the quadratic factors over the whole numbers. That is a fast check on whether factoring is worth attempting.