## What is the Discriminant

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When solving quadratic equations, the **Discriminant** is often used to figure out what the roots are. Even though finding a polynomial's Discriminant is hard, we can use **formulas** to find the Discriminant of **quadratic** and **cubic** equations.

In math, the Discriminant of a polynomial is a function of the **coefficients** of the polynomial. It helps figure out what **kind** of **solutions** a polynomial equation has without having to find them. The word "discriminant" comes from the fact that it tells one solution from another (as equal and unequal; natural and nonreal).

It is often written as \(Δ\) or \(D\). The value of the Discriminant can be an **real **number (i.e., either positive, negative, or \(0\)).

### Formula and the Relationship Between Roots and the Discriminant

The **coefficients** of that polynomial give the Discriminant (\(Δ\) or \(D\)) of a polynomial. The formulas for a cubic equation and a quadratic equation that tell them apart are:

A **quadratic** equation's discriminant formula is: \(ax^2 \ + \ bx \ + \ c \ = \ 0\) is \(Δ\) or \(D \ = \ b^2 \ - \ 4ac\).

A **cubic** equation's discriminant formula is: \(ax^3 \ + \ bx^2 \ + \ cx \ + \ d \ = \ 0\) is \(Δ\) or \(D \ = \ b^2c^2 \ − \ 4ac^3 \ − \ 4b^3d \ − \ 27a^2d^2 \ + \ 18abcd\)

### How Roots and Discriminant Work Together

The values of \(x\) that **work** with the equation \(ax^2 \ + \ bx \ + \ c \ = \ 0\) are called the **roots** of the equation.

We can use the quadratic formula to figure out what they are: \(x \ = \ \frac{-b \ ± \ \sqrt{D}}{2a}\)

Even though we **can't** find the roots just by looking at the Discriminant, we can figure out what **kind** of roots they are in the following way.

**Discriminant > 0:**

If \(D \ > \ 0\), the quadratic equation has **two real** roots. This is because the roots of the equation D > 0 are: \(x \ = \ \frac{-b \ ± \ \sqrt{positive \ number}}{2a}\). And the **square** root of a positive number is **always** a real number.

When the Discriminant of a quadratic equation is **greater** than \(0\), it has two real-number roots that are **different** from each other.

**Discriminant < 0:**

The quadratic equation has two different **complex** roots if \(D\) is **less** than zero. This is because the roots of the equation \(D \ > \ 0\) are: \(x \ = \ \frac{-b \ ± \ \sqrt{negative \ number}}{2a}\). So, you always get an **imaginary** number when you take the square root of a **negative** number.

**Discriminant = 0:**

If \(D \ = \ 0\), the equation has two real roots that are the **same**. This is because the roots of the equation \(D \ = \ 0\) are: \(x \ = \ \frac{-b \ ± \ \sqrt{0}}{2a}\). And the square root would be \(0\). This makes the equation x = b/2a, which is a single number. When the Discriminant of a quadratic equation is 0, it only has one real root.

### Exercises for Use the Quadratic Formula and the Discriminant

**1) **Find the value of the discriminant: \(3x^2 \ - \ 4x \ + \ 1 \ = \ 0\)

**2) **Find the value of the discriminant: \(5x^2 \ + \ 2x \ + \ 3 \ = \ 0\)

**3) **Find the value of the discriminant: \(x^2 \ - \ 3x \ + \ 4 \ = \ 0\)

**4) **Find the value of the discriminant: \(-2x^2 \ + \ 3x \ + \ 4 \ = \ 0\)

**5) **Find the value of the discriminant: \(x^2 \ + \ 2x \ + \ 1 \ = \ 0\)

**6) **Find the value of the discriminant: \(-5x^2 \ - \ 10x \ - \ 5 \ = \ 0\)

**7) **Find the value of the discriminant: \(10x^2 \ + \ 6x \ - \ 2 \ = \ 0\)

**8) **Find the value of the discriminant: \(-6x^2 \ + \ 8x \ + \ 5 \ = \ 0\)

**9) **Find the value of the discriminant: \(5x^2 \ - \ 7x \ + \ 4 \ = \ 0\)

**10) **Find the value of the discriminant: \(-4x^2 \ + \ 6x \ - \ 3 \ = \ 0\)

**1) **Find the value of the discriminant: \(3x^2 \ - \ 4x \ + \ 1 \ = \ 0\)

\(\color{red}{3x^2 \ - \ 4x \ + \ 1 \ = \ 0 \ ⇒ \ a \ = \ 3, \ b \ = \ -4, \ c \ = \ 1}\)

\(\color{red}{Δ \ = \ b^2 \ - \ 4ac \ = \ (-4)^2 \ - \ 4(3)(1) \ = \ 16 \ - \ (12) \ = \ 4}\)

\(\color{red}{Δ \ > \ 0 \ ⇒}\) Two real solutions

**2) **Find the value of the discriminant: \(5x^2 \ + \ 2x \ + \ 3 \ = \ 0\)

\(\color{red}{5x^2 \ + \ 2x \ + \ 3 \ = \ 0 \ ⇒ \ a \ = \ 5, \ b \ = \ 2, \ c \ = \ 3}\)

\(\color{red}{Δ \ = \ b^2 \ - \ 4ac \ = \ (2)^2 \ - \ 4(5)(3) \ = \ 4 \ - \ (60) \ = \ -56}\)

\(\color{red}{Δ \ < \ 0 \ ⇒}\) Two complex solutions

**3) **Find the value of the discriminant: \(x^2 \ - \ 3x \ + \ 4 \ = \ 0\)

\(\color{red}{x^2 \ - \ 3x \ + \ 4 \ = \ 0 \ ⇒ \ a \ = \ 1, \ b \ = \ -3, \ c \ = \ 4}\)

\(\color{red}{Δ \ = \ b^2 \ - \ 4ac \ = \ (-3)^2 \ - \ 4(1)(4) \ = \ 9 \ - \ 16 \ = \ -7}\)

\(\color{red}{Δ \ < \ 0 \ ⇒}\) Two complex solutions

**4) **Find the value of the discriminant: \(-2x^2 \ + \ 3x \ + \ 4 \ = \ 0\)

\(\color{red}{-2x^2 \ + \ 3x \ + \ 4 \ = \ 0 \ ⇒ \ a \ = \ -2, \ b \ = \ 3, \ c \ = \ 4}\)

\(\color{red}{Δ \ = \ b^2 \ - \ 4ac \ = \ 3^2 \ - \ 4(-2)(4) \ = \ 9 \ - \ (-32) \ = \ 41}\)

\(\color{red}{Δ \ > \ 0 \ ⇒}\) Two real solutions

**5) **Find the value of the discriminant: \(x^2 \ + \ 2x \ + \ 1 \ = \ 0\)

\(\color{red}{x^2 \ + \ 2x \ + \ 1 \ = \ 0 \ ⇒ \ a \ = \ 1, \ b \ = \ 2, \ c \ = \ 1}\)

\(\color{red}{Δ \ = \ b^2 \ - \ 4ac \ = \ 2^2 \ - \ 4(1)(1) \ = \ 4 \ - \ 4 \ = \ 0}\)

\(\color{red}{Δ \ = \ 0 \ ⇒}\) One real solutions

**6) **Find the value of the discriminant: \(-5x^2 \ - \ 10x \ - \ 5 \ = \ 0\)

\(\color{red}{x^2 \ + \ 2x \ + \ 1 \ = \ 0 \ ⇒ \ a \ = \ -5, \ b \ = \ -10, \ c \ = \ -5}\)

\(\color{red}{Δ \ = \ b^2 \ - \ 4ac \ = \ (-10)^2 \ - \ 4(-5)(-5) \ = \ 100 \ - \ 100 \ = \ 0}\)

\(\color{red}{Δ \ = \ 0 \ ⇒}\) One real solutions

**7) **Find the value of the discriminant: \(10x^2 \ + \ 6x \ - \ 2 \ = \ 0\)

\(\color{red}{10x^2 \ + \ 6x \ - \ 2 \ = \ 0 \ ⇒ \ a \ = \ 10, \ b \ = \ 6, \ c \ = \ -2}\)

\(\color{red}{Δ \ = \ b^2 \ - \ 4ac \ = \ 6^2 \ - \ 4(10)(-2) \ = \ 36 \ - \ (-80) \ = \ 116}\)

\(\color{red}{Δ \ > \ 0 \ ⇒}\) Two real solutions

**8) **Find the value of the discriminant: \(-6x^2 \ + \ 8x \ + \ 5 \ = \ 0\)

\(\color{red}{-6x^2 \ + \ 8x \ + \ 5 \ = \ 0 \ ⇒ \ a \ = \ -6, \ b \ = \ 8, \ c \ = \ 5}\)

\(\color{red}{Δ \ = \ b^2 \ - \ 4ac \ = \ 8^2 \ - \ 4(-6)(5) \ = \ 64 \ - \ (-120) \ = \ 184}\)

\(\color{red}{Δ \ > \ 0 \ ⇒}\) Two real solutions

**9) **Find the value of the discriminant: \(5x^2 \ - \ 7x \ + \ 4 \ = \ 0\)

\(\color{red}{5x^2 \ - \ 7x \ + \ 4 \ = \ 0 \ ⇒ \ a \ = \ 5, \ b \ = \ -7, \ c \ = \ 4}\)

\(\color{red}{Δ \ = \ b^2 \ - \ 4ac \ = \ (-7)^2 \ - \ 4(5)(4) \ = \ 49 \ - \ 80 \ = \ -31}\)

\(\color{red}{Δ \ < \ 0 \ ⇒}\) Two complex solutions

**10) **Find the value of the discriminant: \(-4x^2 \ + \ 6x \ - \ 3 \ = \ 0\)

\(\color{red}{-4x^2 \ + \ 6x \ - \ 3 \ = \ 0 \ ⇒ \ a \ = \ -4, \ b \ = \ 6, \ c \ = \ -3}\)

\(\color{red}{Δ \ = \ b^2 \ - \ 4ac \ = \ 6^2 \ - \ 4(-4)(-3) \ = \ 36 \ - \ 48 \ = \ -12}\)

\(\color{red}{Δ \ < \ 0 \ ⇒}\) Two complex solutions