Algebra

##### Related Lessons

- Adding and Subtracting Scientific Notation
- Adding Fractions
- Algebra Calculators
- Arithmetic Sequence
- Average Rate of Change
- Change of Base Formula
- Cm to M
- Commutative Property of Addition
- Completing the Square
- Cross Product Calculator
- Determinant Calculator
- Determinant of a Matrix
- Difference of Squares
- Discriminant
- Divisibility Rule for 7
- Dot Product Calculator
- Eigenvalue Calculator
- Eigenvector Calculator
- Equation of a Circle
- Even Numbers
- Exponent Rules
- Factorial Calculator
- Fractional Exponents
- How to Find the Median
- Interval Notation
- Matrix Addition
- Matrix Subtraction
- Midpoint Formula
- Multiplying Negative Numbers
- Negative Exponents
- Odd Numbers
- One to One Function
- Partial Fraction Decomposition Calculator
- Point Slope Form
- Properties of Multiplication
- Rationalize the Denominator
- Rectangular to Polar Calculator
- Reflexive Property
- Round to the Nearest Tenth
- RREF Calculator
- Slope Calculator
- Slope Intercept Form
- Standard Form
- Summation Calculator
- Vertex Form

Tutors/teachers:

Nikkolas

Tutor and Aerospace Engineer

# 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}– 4acThe 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 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) = 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.**

Result :

Worksheet 1

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Cheat sheet

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