Math101 · Independent Ontario learning library
Math101Discriminant
The discriminant predicts the number and type of quadratic roots before the equation is fully solved.
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
| Discriminant | Roots | Graph |
|---|---|---|
| $\Delta>0$ | two distinct real roots | crosses the $x$-axis twice |
| $\Delta=0$ | one repeated real root | touches the $x$-axis at the vertex |
| $\Delta<0$ | no real roots | does not meet the $x$-axis |
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
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?
