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# Discriminant

## What is the Discriminant?

The discriminant is the part of the quadratic formula inside of the square root. For a quadratic ax2 + bx + c = 0, the discriminant is given as:
Discriminant = b2 – 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:
b2 – 4ac > 0 — discriminant is positive — two real roots
b2 – 4ac = 0 — discriminant is zero — repeated real roots
b2 – 4ac < 0 — discriminant is negative — no real roots The 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) = 3x2 + 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 = b2 – 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.

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