Algebra
Quadratic equations and functions
The discriminant
20 practice questions
2 video lessons
Theory + worked examples
Theory
The discriminant is the part under the root in the quadratic formula:
\[D=b^2-4ac.\]
- \(D>0\): two real solutions.
- \(D=0\): one repeated solution.
- \(D<0\): no real solutions.
The discriminant alone reveals the number of real solutions.
Two, one, or no \(x\)-intercepts as \(D\) changes sign.
What the discriminant tells you.
The discriminant:
\[D=b^2-4ac\]
Compute \(D\) to classify before solving.
How to use the discriminant
- Write \(ax^2+bx+c=0\).
- Compute \(D=b^2-4ac\).
- Compare \(D\) with \(0\).
- Read the number of real solutions.
Example 1 — Two real roots
Describe the roots of \(x^2-5x+6=0\).
Solution
Compute \(D\).
| \(D\) | \(=\) | \(25-24=1>0\) |
Two real solutions.
Example 2 — Repeated root
Describe the roots of \(x^2-4x+4=0\).
Solution
Compute \(D\).
| \(D\) | \(=\) | \(16-16=0\) |
One repeated solution.
Example 3 — No real roots
Describe the roots of \(x^2+x+1=0\).
Solution
Compute \(D\).
| \(D\) | \(=\) | \(1-4=-3<0\) |
No real solutions.
Example 4 — What it tells you
Why compute the discriminant?
Solution
It tells the number of real solutions without solving.
Common pitfalls
Watch the signs: \((-5)^2=25\), not \(-25\).
\(D<0\) means no real solutions (in Algebra 1).
Set the equation to \(=0\) first.
Frequently asked questions
What is the discriminant?
\(b^2-4ac\), the part under the root.
What does a positive discriminant mean?
Two real solutions.
What does a zero discriminant mean?
One repeated solution.
What does a negative discriminant mean?
No real solutions.
More in Quadratic equations and functions