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.

Free printable Worksheets

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: $-4x^2 \ + \ 6x \ - \ 3 \ = \ 0$

Answers

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

What is the Discriminant