Discriminant
b² − 4ac tells how many real roots exist.
About this topic
The discriminant is the part of the quadratic formula under the root, b² − 4ac. Its sign tells you the number and type of solutions without solving the whole equation. If it is positive there are two distinct real roots, if zero there is exactly one repeated root, and if negative the roots are a pair of complex numbers with no real solutions.
Geometrically, a positive discriminant means the parabola crosses the x-axis twice, zero means it just touches the axis at its vertex, and negative means it floats above or below the axis. For x² − 4x + 5, the discriminant is 16 − 20 = −4, so there are no real roots. For x² − 6x + 9, it is 36 − 36 = 0, a double root at 3.
If the discriminant is a perfect square, the roots are rational and the quadratic factors over the integers. Engineers use the idea to test whether a design can reach a target, such as whether a projectile can ever reach a given height.
Compute carefully with signs, and remember that a negative discriminant does not mean the equation is wrong, only that the solutions are not real.
Key ideas
- Positive: two real roots
- Zero: one repeated root
- Negative: two complex roots
Quick check
Use cases: where Discriminant helps
- Homework: review the key idea behind Discriminant before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Discriminant without notes.
- Visual learning: play with the interactive lab to see how Discriminant changes when you adjust its inputs.
- Teaching a friend: explain Discriminant in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Discriminant?
b² − 4ac tells how many real roots exist.
What will I learn in Discriminant?
Positive: two real roots. Zero: one repeated root. Negative: two complex roots.
Is Discriminant part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Discriminant lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Discriminant?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.