## What is the Discriminant

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

## Use the Quadratic Formula and the Discriminant Quiz

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