## The Discriminant Video Lesson

## 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 a𝑥 ^{2} + b𝑥 + c, its discriminant is b^{2} – 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, b^{2} – 4ac is represented by the delta symbol, Δ.

The discriminant formula is Δ = b^{2} – 4ac, where a is the coefficient of 𝑥^{2}, b is the coefficient of 𝑥 and c is the constant term of a quadratic.

### For example, calculate the discriminant of y = 𝑥^{2} + 5𝑥 + 2.

We have one 𝑥^{2}. The coefficient of 𝑥^{2} 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, b^{2} – 4ac.

b^{2} = 5^{2} = 25

and 4ac = 4 × 1 × 2 = 8.

b^{2} – 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.

Value of the discriminant | Number of Roots | |

> 0 | Positive | Two |

= 0 | Zero | One |

< 0 | Negative | Zero |

## How to Calculate the Discriminant

**To calculate the discriminant of a quadratic equation, the formula is b ^{2} – 4ac. Substitute the values of a, b and c after reading them from a quadratic equation of the form a𝑥^{2} + b𝑥 + c. For example, for 𝑥^{2} – 3𝑥 + 4, a = 1, b = -3 and c = 4. b^{2} = 9 and 4ac = 16. The discriminant, b^{2} – 4ac = – 7. **

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

- b
^{2}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:**

- Find a, which is the coefficient of 𝑥
^{2}. - Find b, which is the coefficient of 𝑥.
- Find c, which is the constant term.
- Square the b value to find b
^{2}. - Multiply 4 × a × c to find 4ac.
- Use these values to calculate b
^{2}– 4ac.

## Discriminant Calculator

**To use the discriminant calculator:**

- Read the coefficient of 𝑥
^{2}to find ‘a’. - Read the coefficient of 𝑥 to find ‘b’.
- Read the constant term which is ‘c’.

## A Positive Discriminant

**A positive discriminant means that the value of b ^{2} – 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 𝑥^{2} (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 𝑥^{2} (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 𝑥^{2} – 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 b ^{2} – 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 𝑥^{2} (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 𝑥^{2} (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 𝑥^{2} – 4𝑥 + 4 = 0 has a = 1, b = -4 and c = 4.

b^{2} = 16 and 4ac = 16.

b^{2} – 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 b ^{2} – 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 𝑥^{2} (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 𝑥^{2} (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 𝑥^{2} – 3x + 5 =0, a = 1, b = -3 and c = 5.

b^{2} = 9 and 4ac = 20.

Therefore the discriminant b^{2} – 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 a𝑥 ^{3} + b𝑥^{2} + c𝑥 + d is b^{2}c^{2} – 4ac^{3} – 4b^{3}d – 27a^{2}d^{2} + 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. **

Cubic Discriminant Value | Meaning |

Positive discriminant | 3 real roots |

Discriminant of zero | At least one repeated root |

Negative discriminant | 1 real root and 2 complex roots |

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

Using the cubic discriminant formula Δ = b^{2}c^{2} – 4ac^{3} – 4b^{3}d – 27a^{2}d^{2} + 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.