__What is a discriminant?__

A discriminant is a value calculated from a quadratic equation. It use it to 'discriminate' between the roots (or solutions) of a quadratic equation.

A quadratic equation is one of the form: **ax**^{2}** + bx + c**

The discriminant, **D = b**^{2} - 4ac

Note: This is the expression inside the square root of the quadratic formula

There are three cases for the discriminant;

**Case 1:**

**b**^{2}** - 4ac > 0**

If the discriminant is greater than zero, this means that the quadratic equation has **two real, distinct** (different) **roots.**

__Example__

x^{2} - 5x + 2 = 0

a = 1, b = -5, c = 2

Discriminant, D = b^{2} - 4ac

= (-5)^{2 }-^{ }4 * (1) * (2)

= 17

Therefore, there are two real, distinct roots to the quadratic equation

x^{2} - 5x + 2.

**Case 2:**

**b**^{2}** - 4ac < 0**

If the discriminant is greater than zero, this means that the quadratic equation has **no real ****roots.**

__Example__

3x^{2} + 2x + 1 = 0

a = 3, b = 2, c = 1

Discriminant, D = b^{2} - 4ac

= (2)^{2 - }4 * (3) * (1)

= - 8

Therefore, there are no real roots to the quadratic equation 3x^{2} + 2x + 1.

**Case 3:**

**b**^{2}** - 4ac = 0**

If the discriminant is equal to zero, this means that the quadratic equation has** two real, identical roots**.

__Example__

x^{2} + 2x + 1 = 0

a = 1, b = 2, c = 1

Discriminant, D = b^{2} - 4ac

= (2)^{2 }- 4 * (1) * (1)

= 0

Therefore, there are two real, identical roots to the quadratic equation x^{2} + 2x + 1.

**Summary**

**Quadratic equation is ax ^{2} + bx + c**

**Determinant D = b ^{2} - 4ac**

**D > 0 means two real, distinct roots.**

**D = 0 means two real, identical roots/**

**D < 0 means no real roots.**

Now try these, (take care with minus signs)

__Questions__

Q1. x^{2} - 7x + 2 = 0

Q2. - 3x^{2} + 2x - 1 = 0

Q3. 9x^{2} - 12x + 4 = 0

Q4. - x^{2} + x + 1 = 0

__Answers__

Q1. D = 41, means two real, distinct roots.

Q2. D = -16, means no real roots.

Q3. D = 0, means two real, identical roots.

Q4. D = 5, means two real, distinct roots.

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