The discriminant is b² − 4ac — the part under the square root in the quadratic formula.
Step 1: Let's Learn
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Positive
A positive discriminant gives two distinct real roots, because the root is a real non-zero number.
Zero
A zero discriminant gives exactly one repeated root, where the parabola just touches the axis.
Negative
A negative discriminant gives no real roots, because no real number squares to a negative.
The part under the root
The discriminant is b² − 4ac, the expression inside the square root in the quadratic formula. Its sign alone determines how many real roots the equation has, without computing the roots themselves.
Positive, zero, negative
Positive gives two distinct real roots, because you add and subtract a nonzero root. Zero gives one repeated root. Negative gives no real roots, since no real number squares to a negative.
What each case looks like
Two roots means the parabola crosses the axis twice; zero means it touches at the vertex; negative means it floats entirely above or below. The discriminant is an algebraic reading of a geometric fact.
Why compute it first
Checking the discriminant before solving tells you what kind of answer to expect, and whether a real solution exists at all. In modelling problems, "no real roots" is often the meaningful answer — the projectile never reaches that height.
Step 2: Try It Yourself
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Step 3: Watch an Example
One step at a time.
Watch Priya Count Without Solving
Priya examines x² + 2x + 5.
- Step 1
Here a = 1, b = 2, and c = 5.
Step 4: Your Turn
Practice makes it stick.
The Value
Problem 1 of 2
a = 1, b = 6, c = 5. What is the discriminant?
The Count
Problem 2 of 2
Discriminant 16. How many real roots?
Count the Roots
1 of 8
a = 1, b = 4, c = 4. Discriminant?
2 of 8
Discriminant 0. How many real roots?
3 of 8
a = 1, b = 1, c = 5. Discriminant?
4 of 8
Discriminant −19. How many real roots?
5 of 8
a = 2, b = 5, c = 2. Discriminant?
6 of 8
Sort each discriminant by the number of real roots it gives.
Tap something to move it.
- Empty
- Empty
- Empty
7 of 8
A parabola touching the axis exactly once. Discriminant?
8 of 8
a = 1, b = 0, c = −9. Discriminant?
Step 5: Quick Check
Show what you know.
Question 1 of 1
a = 1, b = 2, c = 1. Discriminant?
What You Learned
- The discriminant is b² − 4ac, the part under the root.
- Positive gives two roots, zero gives one, negative gives none.
- It answers how many before any solving is done.