# A Complete Guide to the Discriminant of Quadratic

## What is the Discriminant of a Quadratic Equation?

The discriminant is the part of the quadratic formula found within the square root. For a quadratic of the form a𝑥2 + b𝑥 + c, its discriminant is b2 – 4ac. A quadratic equation has 2, 1 or 0 solutions depending if the value of the discriminant is positive, zero or negative respectively.

The discriminant, b2 – 4ac is represented by the delta symbol, Δ.

The discriminant formula is Δ = b2 – 4ac, where a is the coefficient of 𝑥2, b is the coefficient of 𝑥 and c is the constant term of a quadratic.

### For example, calculate the discriminant of y = 𝑥2 + 5𝑥 + 2.

We have one 𝑥2. The coefficient of 𝑥2 is 1. Therefore a = 1.

The coefficient of 𝑥 is 5. Therefore b = 5.

The constant term is 2. Therefore c = 2.

We substitute the values of a = 1, b = 5 and c = 2 into the formula for the discriminant, b2 – 4ac.

b2 = 52 = 25

and 4ac = 4 × 1 × 2 = 8.

b2 – 4ac becomes 28 – 8 = 17. The discriminant is 17.

The discriminant is important because it tells us how many solutions any quadratic equation has.

• If the discriminant is a positive number, there will be 2 solutions.
• If the discriminant is zero, there will be 1 solution.
• If the discriminant is a negative number, there will not be any real solutions.

The number of solutions to a quadratic equation tells us the number of roots of the quadratic equation. The roots of a quadratic equation are the locations where the quadratic graph crosses the 𝑥-axis. They are the 𝑥-axis intercepts.

The following table shows the number of roots for a positive, negative or zero discriminant.

## How to Calculate the Discriminant

To calculate the discriminant of a quadratic equation, the formula is b2 – 4ac. Substitute the values of a, b and c after reading them from a quadratic equation of the form a𝑥2 + b𝑥 + c. For example, for 𝑥2 – 3𝑥 + 4, a = 1, b = -3 and c = 4. b2 = 9 and 4ac = 16. The discriminant, b2 – 4ac = – 7.

When calculating the discriminant it is important to consider these key points:

• b2 is always positive. When we square a negative number, it gives us a positive result.
• If 4ac is negative then we need to perform an addition. When we subtract a negative number, an addition takes place.
To find the discriminant:

1. Find a, which is the coefficient of 𝑥2.
2. Find b, which is the coefficient of 𝑥.
3. Find c, which is the constant term.
4. Square the b value to find b2.
5. Multiply 4 × a × c to find 4ac.
6. Use these values to calculate b2 – 4ac.

## Discriminant Calculator

To use the discriminant calculator:

1. Read the coefficient of 𝑥2 to find ‘a’.
2. Read the coefficient of 𝑥 to find ‘b’.
3. Read the constant term which is ‘c’.

## A Positive Discriminant

A positive discriminant means that the value of b2 – 4ac is greater than zero. A quadratic equation with a positive discriminant has exactly two solutions, which means that it has two 𝑥-axis intercepts. This means that the quadratic has 2 roots.

A positive value of the discriminant means that the graph of the quadratic equation must pass through the 𝑥-axis twice.

If the quadratic has a positive coefficient of 𝑥2 (a > 0), the graph is concave up and the minimum point will be below the 𝑥-axis as shown in the left image below.

If the quadratic has a negative coefficient of 𝑥2 (a < 0), the graph is concave down and the minimum point will be above the 𝑥-axis as shown in the right image below.

A positive value of the discriminant tells us that a quadratic has two unique solutions.

For example, the quadratic 𝑥2 – 4𝑥 + 3 = 0 has two solutions: 𝑥 = 3 and 𝑥 = 1.

This means that the quadratic equation crosses the 𝑥-axis at 𝑥 = 1 and 𝑥 = 3.

The discriminant value is 4, which is a positive number.

The square root of a positive number has both a positive and negative answer. Therefore the quadratic formula provides 2 different solutions.

If the discriminant is a perfect square, then the solutions to the quadratic equation are rational. This means that the solutions will be integers or can be written as fractions. If a discriminant is not a perfect square, the 2 solutions will be irrational. This is because the square root of this discriminant will be a surd.

## A Discriminant of Zero

A discriminant of zero means that the value of b2 – 4ac is equal to zero. A quadratic equation with a discriminant of zero has exactly one solution. This means that the graph of the quadratic just touches the 𝑥-axis at its minimum or maximum point.

A discriminant equal to zero means that the graph of the quadratic equation must touch the 𝑥-axis once. It cannot pass through the 𝑥-axis. Instead it just touches it at its one root.

If the quadratic has a positive coefficient of 𝑥2 (a > 0), the graph is concave up and the minimum point will touch 𝑥-axis as shown in the left image below.

If the quadratic has a negative coefficient of 𝑥2 (a < 0), the graph is concave down and the maximum point will touch the 𝑥-axis as shown in the right image below.

For a quadratic equation with a discriminant of zero, there will be exactly one solution. This is because the square root of the discriminant is taken as part of the quadratic formula. The square root of 0 is 0.

This means that we add or subtract 0 in the calculation, which causes the two answers to be the same.

The solution to a quadratic equation with a discriminant of zero is called a repeated root. This is because the same solution appears twice.

For example, the quadratic 𝑥2 – 4𝑥 + 4 = 0 has a = 1, b = -4 and c = 4.

b2 = 16 and 4ac = 16.

b2 – 4ac = 0 and the one repeated root of the quadratic equation is 𝑥 = 2.

## A Negative Discriminant

A negative discriminant means that the value of b2 – 4ac is less than zero. When using the quadratic formula the square root of a negative cannot be found. There are no real solutions to the quadratic equation. The two solutions are complex and cannot be seen on a graph.

A negative discriminant means that the graph of the quadratic equation does not touch the 𝑥-axis.

If the quadratic has a positive coefficient of 𝑥2 (a > 0), the graph is concave up and the entirety of the graph is above the 𝑥-axis. All outputs of the graph are positive.

If the quadratic has a negative coefficient of 𝑥2 (a < 0), the graph is concave down and the entirety of the graph is below the 𝑥-axis. All outputs of the graph are negative.

For example, in the quadratic equation 𝑥2 – 3x + 5 =0, a = 1, b = -3 and c = 5.

b2 = 9 and 4ac = 20.

Therefore the discriminant b2 – 4ac = -11.

The square root of a negative number cannot be found and so, no real solutions can be found since we cannot complete the calculation.

The square root of -11 must be written in terms of the imaginary unit, i.

Therefore it is written as √11 i. This complex solution can be used.

## The Discriminant from a Graph

The number of 𝑥-axis intercepts indicates the value of the discriminant:

• A graph with 2 𝑥-axis intercepts has a positive discriminant.
• A graph that touches the 𝑥-axis once has a discriminant of zero.
• A graph that does not touch the 𝑥-axis has a negative discriminant.

## The Discriminant of a Cubic Equation

The discriminant of a cubic equation a𝑥3 + b𝑥2 + c𝑥 + d is b2c2 – 4ac3 – 4b3d – 27a2d2 + 18abcd. If the discriminant is positive, the cubic has 3 real roots. If it is negative, the cubic has one real root and two complex conjugate roots. If it is zero, at least two of the roots are equal.

For example for the cubic 5𝑥3 + 2𝑥2 – 3𝑥 + 1, a = 5, b = 2, c = -3 and d = 1.

Using the cubic discriminant formula Δ = b2c2 – 4ac3 – 4b3d – 27a2d2 + 18abcd.

Δ = -671.

The cubic discriminant is negative and so, this cubic has 1 real root and 2 complex roots.

This means that its graph will only intersect the 𝑥-axis once.