Lesson Explainer: Discriminants of Quadratics Mathematics

In this explainer, we will learn how to find the discriminant of a quadratic equation and use it to determine the number and type of its roots (solution) without solving it.

We recall that a general quadratic equation takes the form

π‘Žπ‘₯+𝑏π‘₯+𝑐=0,()1

where π‘Ž, 𝑏, and 𝑐 are real numbers and π‘₯ is the variable to be found. For this equation to be a quadratic, we require that π‘Žβ‰ 0, but we do not enforce the same restriction on 𝑏 or 𝑐. This equation is β€œsolved” when a value of π‘₯ is found such that equation (1) is true. For a quadratic equation, there might be a maximum of two real solutions to equation (1), as opposed to the single real solution of a linear equation. To be more specific, for any quadratic equation with real coefficients, there will either be 0, 1, or 2 real solutions.

It is well known that the solution of a quadratic equation is given by the quadratic root formula,

π‘₯=βˆ’π‘Β±βˆšπ‘βˆ’4π‘Žπ‘2π‘Ž,()2

with the two possible solutions represented by the Β± sign. The validity of either of these two solutions can be checked algebraically by substituting either expression in equation (2) into equation (1).

The quadratic root formula alludes to the fact that there might be either 0, 1, or 2 real solutions to the general quadratic equation because the Β± symbol suggests that there are two possible calculations to find π‘₯. If the expression inside the square root term is positive, then there is no problem with finding the solution. However, if the expression inside the square root term is negative, then we will be attempting to take the square root of a negative number, for which there are no solutions in the real numbers. Finally, if the expression inside the root symbol is zero, then both calculations will be equal, so we will only have one root. Hence, the number of real solutions is determined by the sign of the expression π‘βˆ’4π‘Žπ‘οŠ¨, which is known as the discriminant.

Definition: The Discriminant of a Quadratic

Consider a quadratic equation π‘Žπ‘₯+𝑏π‘₯+𝑐=0, where π‘Ž, 𝑏, and 𝑐 are real numbers and π‘Žβ‰ 0. Then, the β€œdiscriminant” of the quadratic is denoted Ξ”=π‘βˆ’4π‘Žπ‘.

If Ξ” is positive, then there are two real solutions to the quadratic equation. If Ξ”=0, then there is one (repeated) real solution. And if Ξ” is negative, then there are no real solutions.

Having now made this definition, we can see how it is possible to write the quadratic root formula in terms of the discriminant as π‘₯=βˆ’π‘Β±βˆšΞ”2π‘Ž, which further clarifies the connection to the number of solutions of the quadratic. We will demonstrate this idea by way of example, by considering the quadratic equation 4π‘₯βˆ’1+4π‘₯=0. It will be a useful paradigm to think instead in terms of the function 𝑓(π‘₯)=4π‘₯+4π‘₯βˆ’1, and then ask for the values of π‘₯ that give 𝑓(π‘₯)=0. In other words, we are then trying to find the roots of the function 𝑓(π‘₯). We begin by plotting the function, as shown below, that reveals that there are two roots, with one being negative and the other being positive.

We will now confirm this by referring to our definition of the discriminant. To solve the equation 4π‘₯βˆ’1+4π‘₯=0, we should first note that this is a quadratic equation in π‘₯ with coefficients π‘Ž=4, 𝑏=4, and 𝑐=βˆ’1. The discriminant is then calculated as Ξ”=π‘βˆ’4π‘Žπ‘=4βˆ’4Γ—4Γ—(βˆ’1)=32.

Therefore, we have Ξ”>0, which, according to the definition above, means that there are two real solutions. This was also suggested by the graph that we plotted above. We can then use the quadratic to directly calculate these roots as follows: π‘₯=βˆ’π‘Β±βˆšΞ”2π‘Ž=βˆ’4±√322Γ—4=βˆ’1±√22.

We can check that these two real solutions correspond numerically to those shown in the plot above.

Let’s consider another example, this time for the quadratic 4π‘₯+1+4π‘₯=0, meaning that we set 𝑔(π‘₯)=4π‘₯+4π‘₯+1, which is a quadratic function where π‘Ž=4, 𝑏=4, and 𝑐=1. Compared to the function 𝑓(π‘₯), the function 𝑔(π‘₯) will have exactly the same shape after a translation by two units in the positive vertical direction. The graph of this function is as follows:

It appears as though there is only one real solution to this equation at π‘₯=βˆ’12, which we will show is exactly the case by use of the discriminant. We calculate the discriminant as follows: Ξ”=π‘βˆ’4π‘Žπ‘=4βˆ’4Γ—4Γ—1=0.

The fact that Ξ”=0 means that there is one real (repeated) root, as appears to be the case from the graph plotted above. Then, the quadratic formula is used to calculate the solutions as π‘₯=βˆ’π‘Β±βˆšΞ”2π‘Ž=βˆ’4±√02Γ—4=βˆ’12.

In this case, the Β± term is irrelevant because adding zero is the same as subtracting zero. Unlike the previous scenario, there is no need to complete further calculations in order to the find the (repeated) real root.

The final example that we shall give will be for a quadratic equation where there are no real solutions. We will take the previous example and modify it slightly to give the quadratic equation 3+4π‘₯+4π‘₯=0. To aid in our process, we will define the function β„Ž(π‘₯)=4π‘₯+4π‘₯+3, which is a quadratic function with π‘Ž=4, 𝑏=4, and 𝑐=3. This graph of β„Ž(π‘₯) is the same as the graph of 𝑔(π‘₯) translated two units in the positive vertical direction, the result of which is shown below.

It appears from this graph that there are no real solutions to the quadratic equation, which we can show by calculating the discriminant, as follows: Ξ”=π‘βˆ’4π‘Žπ‘=4βˆ’4Γ—4Γ—3=βˆ’32.

This shows that Ξ”<0, which means that there are no real solutions, thereby confirming our prediction from having plotted the graph. Attempting to use the quadratic root formula will give the following working: π‘₯=βˆ’π‘Β±βˆšΞ”2π‘Ž=βˆ’4Β±βˆšβˆ’322Γ—4.

This working shows that we are trying to calculate the square root of a negative number, which does not give a result that is a real number. This means that there are no real solutions to the original quadratic equation, as predicted by the value of the discriminant. In this situation, understanding the solutions will require an understanding of imaginary and complex numbers, which are outside the scope of this explainer.

Now let’s see some examples of how the discriminant is used to determine the number of real roots of a quadratic equation.

Example 1: Using the Sign of the Discriminant to Determine the Number of Complex Roots of a Quadratic Equation

How many nonreal roots will a quadratic equation have if its discriminant is negative?

Answer

We recall that if we have the quadratic equation π‘Žπ‘₯+𝑏π‘₯+𝑐=0, where π‘Ž, 𝑏, and 𝑐 are real numbers and π‘Žβ‰ 0, then we know that the quadratic root formula is π‘₯=βˆ’π‘Β±βˆšπ‘βˆ’4π‘Žπ‘2π‘Ž, which gives us the roots of the quadratic. The discriminant is defined as Ξ”=π‘βˆ’4π‘Žπ‘οŠ¨, which allows the quadratic root formula to instead be written as π‘₯=βˆ’π‘Β±βˆšΞ”2π‘Ž.

If the discriminant is negative, then we will be attempting to calculate the square root of a negative number, which has no solutions in the real numbers. This means that there are zero real solutions to the given quadratic equation, which means that there must be two nonreal roots.

Example 2: Using the Sign of the Discriminant to Determine the Number of Complex Roots of a Quadratic Equation

Which is the correct condition for the quadratic equation π‘Žπ‘₯+𝑏π‘₯+𝑐=0 with real coefficients to have no nonreal roots?

  1. The discriminant π‘βˆ’4π‘Žπ‘οŠ¨ is positive.
  2. The discriminant π‘βˆ’4π‘Žπ‘οŠ¨ is equal to zero.
  3. The discriminant π‘βˆ’4π‘Žπ‘οŠ¨ is negative.
  4. The discriminant π‘βˆ’4π‘Žπ‘οŠ¨ is nonnegative.
  5. The discriminant π‘βˆ’4π‘Žπ‘οŠ¨ is an integer.

Answer

When working with a quadratic, we recall that the sign of the discriminant tells us the number of real roots. There are three possibilities for the number of real roots:

  • Two real roots, when Ξ”=π‘βˆ’4π‘Žπ‘>0
  • One real (repeated) root, when Ξ”=π‘βˆ’4π‘Žπ‘=0
  • No real roots, when Ξ”=π‘βˆ’4π‘Žπ‘<0

We are told that we are looking for there to be no nonreal roots to the given quadratic. This means that there must be at least one real solution to the quadratic equation, with the maximum being two. For there to be one real solution, we require that the discriminant be equal to zero, and for there to be two real solutions, we require that the discriminant be positive. For either of these conditions to be satisfied, we require that the discriminant be greater than or equal to zero. This corresponds to option D from the list above, meaning that the discriminant must be nonnegative.

The two examples above demonstrate how the number of roots can be classified simply using the discriminant. When trying to find the precise roots of a quadratic equation, a useful preemptive step is therefore to calculate the discriminant and then use this to understand the number of roots before we calculate them. For example, if the discriminant of a quadratic is negative, then there are no real roots and hence, there is no need to employ the quadratic formula to find these. We will give an example of this in the next question.

Example 3: Finding the Discriminant of a Quadratic Equation and Using It to Determine the Number of Real Roots

  1. Find the discriminant of the quadratic equation 2π‘₯+3π‘₯+4=0.
  2. How many real roots does the equation 2π‘₯+3π‘₯+4=0 have?
  3. Hence, decide how many times the graph of 𝑦=2π‘₯+3π‘₯+4 will cross the π‘₯-axis.

Answer

Part 1

We will begin by noting that the above quadratic equation can be classified in the normal way by writing the coefficients as π‘Ž=2, 𝑏=3, and 𝑐=4. We recall that the discriminant of a quadratic is Ξ”=π‘βˆ’4π‘Žπ‘οŠ¨, which we can calculate for this quadratic as follows: Ξ”=π‘βˆ’4π‘Žπ‘=3βˆ’4Γ—2Γ—4=βˆ’23.

Part 2

We recall that the sign of the discriminant of a quadratic tells us the number of real roots that the quadratic has. In particular, if its sign is negative, then there are no real roots. Given that Ξ”<0, this means that there are no real roots to the given quadratic equation, so the answer is zero real roots.

Part 3

A function has a root when the graph of that function crosses the π‘₯-axis. Given that this function has no real roots, this implies that the graph of the function does not cross the π‘₯-axis. This can be confirmed graphically using the plot below of the function 𝑓(π‘₯)=2π‘₯+3π‘₯+4.

We see that the graph of the function will never cross the π‘₯-axis, as predicted.

We have already seen that the quadratic root formula can be phrased in terms of the discriminant as π‘₯=βˆ’π‘Β±βˆšΞ”2π‘Ž.

Before calculating the roots of a quadratic, we will need to calculate the square root of the discriminant Ξ”. This means that if Ξ” is a square number, then the square root will return an integer. Providing that π‘Ž and 𝑏 are both rational, in this particular case, the values of π‘₯ will therefore be rational. However, it will generally be the case that Ξ” is not a square number, which means that the square root of this value will be an irrational number. When this is the case, it will imply that the values of π‘₯ will be irrational, since they will be the combination of an irrational number and two rational numbers using addition and division. Note that this property only holds under the assumption that π‘Ž and 𝑏 are both rational. If they are not both rational, then we will need to consider the question slightly more delicately, as we will see in the following example.

Example 4: Determining Whether the Roots of a Quadratic Equation Are Rational or Not by Using the Discriminant

Determine whether the roots of the equation π‘₯βˆ’βˆš5π‘₯βˆ’1=0 are rational or not without solving it.

Answer

We set the coefficients of this quadratic equation in the standard way, by fixing π‘Ž=1, 𝑏=βˆ’βˆš5, and 𝑐=βˆ’1. We recall that the discriminant of a quadratic π‘Žπ‘₯+𝑏π‘₯+𝑐=0 is Ξ”=π‘βˆ’4π‘Žπ‘οŠ¨, and the quadratic formula tells us the roots of this quadratic are π‘₯=βˆ’π‘Β±βˆšΞ”2π‘Ž.

We can then calculate the discriminant of the quadratic as follows: Ξ”=π‘βˆ’4π‘Žπ‘=ο€»βˆ’βˆš5ο‡βˆ’4Γ—1Γ—(βˆ’1)=5+4=9.

We know that since this is positive, there are two real roots. We can also see that, in the quadratic formula, 2π‘Ž is rational and βˆšΞ”=√9=3 is also rational. However, βˆ’π‘ is irrational; hence, the roots will be irrational.

In our next example, we will investigate the behavior of quadratics by considering the coefficient 𝑐 to be a parameter.

Example 5: Finding the Interval to Which a Variable in a Quadratic Equation Belongs given the Type of Its Roots

Given that the roots of the equation 4π‘₯βˆ’12π‘₯+π‘˜=0 are real and different, find the interval that contains π‘˜.

Answer

We begin by treating this quadratic in the normal fashion. We label the parameters as π‘Ž=4, 𝑏=βˆ’12, and 𝑐=π‘˜. We recall that the sign of the discriminant of a quadratic π‘Žπ‘₯+𝑏π‘₯+𝑐=0 gives us the number of roots of the quadratic. In this question, we want two distinct real roots, which occurs when the discriminant is positive. We calculate the discriminant Ξ” as follows: Ξ”=π‘βˆ’4π‘Žπ‘=(βˆ’12)βˆ’4Γ—4Γ—π‘˜=144βˆ’16π‘˜=16(9βˆ’π‘˜).

The question asked us to find all possible values of π‘˜ that ensure that the roots of the quadratic are real and different. In other words, we are asked to find the possible values of π‘˜ such that there are two real roots, meaning that Ξ”>0. By looking at the factored expression in the equation above, we deduce that Ξ”>0 if 9βˆ’π‘˜>0. Rearranging this gives π‘˜<9. Note that, we cannot allow the situation π‘˜=9, as this will suggest that there is a single (repeated) real root. Given that we require π‘˜<9, we can alternatively express this as π‘˜βˆˆ]βˆ’βˆž,9[.

Key Points

  • A quadratic equation in the variable π‘₯ is defined as π‘Žπ‘₯+𝑏π‘₯+𝑐=0, where π‘Ž, 𝑏, and 𝑐 are all real numbers and π‘Žβ‰ 0.
  • The roots of a quadratic equation are given by the quadratic root formula as follows: π‘₯=βˆ’π‘Β±βˆšπ‘βˆ’4π‘Žπ‘2π‘Ž.
  • The discriminant of the quadratic π‘Žπ‘₯+𝑏π‘₯+𝑐=0 is given by Ξ”=π‘βˆ’4π‘Žπ‘οŠ¨. The sign of Ξ” tells us the number of real roots, as follows:
    • If Ξ”>0, then there are two distinct real solutions.
    • If Ξ”=0, then there is one repeated real solution.
    • If Ξ”<0, then there are no real solutions.
  • It can be helpful to rephrase the quadratic root formula as π‘₯=βˆ’π‘Β±βˆšΞ”2π‘Ž.
  • Assuming that π‘Ž, 𝑏, and 𝑐 are all rational, then the values of π‘₯ will be rational if, and only if, Ξ” is a square number.

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