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# Discriminant

#### Lesson Contents

## What is the Discriminant?

**The discriminant is the part of the quadratic formula inside of the square root.** For a quadratic *ax ^{2} + bx + c = 0*, the discriminant is given as:

**Discriminant = b ^{2} – 4ac**

The discriminant tells us how many roots a quadratic has. The relationship between result of discriminant and number of roots a quadratic has is given as:

**b ^{2} – 4ac > 0 — discriminant is positive — two real roots**

**b**

^{2}– 4ac = 0 — discriminant is zero — repeated real roots**b**

^{2}– 4ac < 0 — discriminant is negative — no real rootsThe quadratic formula with the discriminant boxed in red.

### Why we use the Discriminant

The advantage of using the discriminant instead of just solving the full quadratic formula is that **we get a quicker idea of how many roots the quadratic will have.** Additionally, we get a quick and accurate idea of the type of roots.

For example, let’s look at a quadratic whose discriminant is zero. A discriminant of zero means there are repeated real roots. If we had graphed the quadratic, it may appear as only having a single root (location where the curve crosses the x-axis). Thanks to the discriminant, we know that there are actually repeated roots in the graph.

### Example Problem

Use the discriminant to determine the number of and type of roots of the function f(x) = 3x^{2} + 4x + 1.

Solution:

1.) Since the function is a standard quadratic, let’s set a = 3, b = 4, and c = 1

2.) Plugging the constants into the discriminant, we get:

Discriminant = b^{2} – 4ac

Discriminant = (4)^{2} – 4(3)(1) = 16 – 12 = 4.

3.) The discriminant is 4, a positive number. **Therefore, the given quadratic function has two real roots.**