What is the Discriminant

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

 

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