Discriminant

The expression under the quadratic formula's square root tells us how a parabola meets the $x$-axis.

Definition

For a quadratic equation

the discriminant is

The Greek letter $\Delta$ is read “delta.” The discriminant is not the full solution; it classifies what kind of solutions are possible.

Why it matters

The quadratic formula is

Only the expression beneath the square root can make the real-number calculation positive, zero, or impossible. Its sign determines how many distinct real roots exist.

Three real-number cases

Over complex numbers, a negative discriminant gives two conjugate complex roots rather than “no solutions.” The intended number system matters.

Worked example: classify before solving

Rational and irrational roots

When $a$, $b$, and $c$ are integers and $\Delta>0$:

• if $\Delta$ is a perfect square, both real roots are rational;
• if $\Delta$ is not a perfect square, the roots are irrational.

For $x^2-2x-1=0$, $\Delta=8>0$ but is not a perfect square, so the two roots $1\pm\sqrt2$ are irrational.

A repeated root

Consider $x^2-6x+9=0$:

The quadratic is $(x-3)^2=0$, so $x=3$ is a repeated root. The parabola's vertex is $(3,0)$ and merely touches the $x$-axis.

A negative discriminant

For $x^2+4x+8=0$,

There are no real roots. Completing the square gives $(x+2)^2+4=0$, whose left side is always positive for real $x$. Over complex numbers, the roots are $-2\pm2i$.

Determining an unknown parameter

The discriminant can impose a condition without solving the full quadratic. Suppose

has exactly one real root. Set the discriminant equal to zero:

Both parameter values produce a perfect-square trinomial.

Geometry and modelling

If a projectile's height equation is set equal to a target height, the discriminant can show whether the object reaches that level twice, touches it once at its maximum, or never reaches it. In analytic geometry it can determine whether a line intersects, is tangent to, or misses a parabola.

Classification can therefore answer a question even when exact roots are not requested.

Common mistakes

Using the wrong sign for $b$. Substitute coefficients with their signs and use parentheses.

Forgetting that $b$ is squared. If $b=-5$, then $b^2=25$.

Calling $\Delta=0$ “no roots.” It gives one distinct repeated real root.

Stopping at “two roots” when type is requested. Check whether a positive discriminant is a perfect square.

Using an equation not equal to zero. Identify $a$, $b$, and $c$ only after writing standard form.

Quick self-check

• Is the equation in $ax^2+bx+c=0$ form?
• Did I include coefficient signs correctly?
• Is $b^2-4ac$ evaluated with parentheses?
• Does the sign of $\Delta$ match the graph's intercept behaviour?
• If $\Delta>0$, is it a perfect square?

Related topics

• Quadratic Formula
• Completing the Square
• Vertex Form
• Roots and Zeros