In this explainer, we will learn how to solve quadratic equations whose roots are complex numbers.
The introduction of complex numbers opens up the possibility of finding solutions to equations which we were formerly unable to solve. For example, formerly, when we came across an equation that required us to take the square root of a negative number, we were unable to solve it and rightly concluded there were no real solutions. However, using complex numbers, we can gain additional insight by exploring the complex solutions to these equations. We begin by looking at an example of an equation we are unable to solve if we restrict ourselves to dealing solely with real numbers.
Example 1: Solving Equations with Complex Numbers
Solve the equation .
We start by gathering our like terms:
Dividing both sides by 5, we isolate on the left of the equation as follows:
Taking the square root of both sides remembering that we can take both the positive and the negative roots, we get
We recall the property of complex numbers, for a positive number , , we can rewrite this as
As we observed in the previous example, the methods we apply to solve equations with real solutions can often be directly applied to equations with complex solutions. When dealing with quadratic equations, methods such as factoring and completing the square can equally be applied to equations with complex solutions. In particular, we can also use the quadratic formula.
Formula: Quadratic Formula
For a quadratic equation with , the roots are given by
Note that these roots are sometimes referred to as and .
Using the quadratic formula, we are able to solve any quadratic equation including the ones with complex solutions. When using the quadratic formula, we come across three distinct cases. To distinguish between them, we introduce the idea of the discriminant.
The discriminant of a quadratic equation is defined as . Often is used to denote the discriminant.
Using the discriminant, we identify the three different cases of quadratic equations as follows:
- Positive discriminant: , two real roots,
- Zero discriminant: , one repeated real root,
- Negative discriminant: , no real roots.
Let us consider the graphs below depicting these three cases.
We will primarily focus on graph (3), where the quadratic equation has no real roots. The introduction of complex numbers enables us to restate this as the case were we have complex roots. Although real numbers are also complex numbers, when we say that a quadratic equation has complex roots, we specifically refer to the case when the roots are nonreal complex numbers. In this explainer, we will explore this case and the properties of complex roots.
Let us consider an example where we will use the quadratic formula to obtain complex roots of a quadratic equation.
Example 2: Solving Quadratic Equations with Complex Roots
Solve the quadratic equation .
Recall the quadratic formula for solving the quadratic equation :
The given quadratic equation has the values , , and . Substituting these values into the quadratic formula, we have
Simplifying, we get
Recalling the property of complex numbers for a positive number , , we can rewrite this as
Hence, we have two solutions for the quadratic equation:
In the previous example, we observed that the quadratic equation had two complex solutions. When we examine the solutions closely, we can notice that the two complex solutions are conjugates of each other. The fact that the roots of this equation are a complex conjugate pair is no accident. In fact, it is true for any quadratic equation with real coefficients which has complex solutions.
In the next example, we will examine this fact in detail.
Example 3: Conditions on the Roots of Quadratic Equations
If the discriminant of a quadratic equation with real coefficients is negative, will its complex roots be a conjugate pair?
Recall that the discriminant of a quadratic equation is the expression . If the discriminant is negative, then we know that the quadratic equation will have complex roots. Let us examine whether or not these complex roots must be complex conjugates.
Using the properties of the complex conjugate, we will provide a proof of this theorem. For a quadratic function , let be a complex root. Now, we consider
From the properties of the complex conjugate, we know that for any two complex numbers ; hence, . Therefore, we have
Given that any real number is equal to its complex conjugate and that , , and are real numbers, we can rewrite this as
We also know from the properties of the complex conjugate that for any two complex numbers ; hence,
However, since is a root, we know that . Therefore, . Hence, we have shown that is also a root of .
We can summarize the result of the previous example as follows.
Theorem: Conjugate Root Theorem for Quadratic Equations
The complex roots of a quadratic equation with real coefficients occur in complex conjugate pairs. Hence, if (where ) is a root of a quadratic equation with real coefficients, then is also a root.
Both this theorem and its proof can be generalized to any polynomial with real coefficients. In the remainder of this explainer, we will apply this theorem to a number of examples.
Example 4: Complex Roots of Quadratic Equations
The complex numbers and , where , , , and are real numbers, are the roots of a quadratic equation with real coefficients. Given that , what conditions, if any, must , , , and satisfy?
Since , we know that is a complex root to a quadratic equation with real coefficients. We recall the conjugate root theorem, which states that the complex roots of a quadratic equation with real coefficients occur in complex conjugate pairs. Furthermore, since a quadratic equation only has two roots, must be the conjugate of . Hence,
We remember that complex numbers are equal to each other if their real parts are equal and their imaginary parts are equal.
Therefore, , , , and must satisfy and .
We can also use our knowledge of the roots of quadratic equations with real coefficients to reconstruct an equation given one of its complex roots as the next example will demonstrate.
Example 5: Reconstructing a Quadratic Equation from a Complex Root
Find the quadratic equation with real coefficients and that has as one of its roots.
In this example, we are given that a quadratic equation with real coefficients has a complex root. We recall the conjugate root theorem, which states that the complex roots of a quadratic equation with real coefficients occur in complex conjugate pairs. Hence, its roots will be and . Since the coefficient of is equal to 1, we can write the equation as
Treating and as single terms, we can multiply through the parenthesis to obtain
Now, we can multiply through the parenthesis in and simplify:
Using , we have
In the previous example, we were able to find a quadratic equation with real coefficients when we were given one complex root. More generally, if we have a quadratic equation with roots and , we can write the equation as
Multiplying through the parenthesis, we obtain
This is true for any quadratic equation whether the roots are real or complex. However, if and are conjugate pairs, we can write , which gives
We recall the properties of complex conjugates and , where is the modulus of the complex number . Hence, if we know that is a complex root of a quadratic equation with real coefficients, one possible quadratic equation is
By applying this knowledge, it is possible to significantly simplify the calculation presented in the previous example.
Let us finish by recapping a few important concepts from this explainer.
- Using the same methods developed to solve quadratic equations with real roots, we can solve quadratic equations with complex roots. Although a real number is technically also a complex number, when we refer to complex roots of a quadratic equation, we mean that the roots are nonreal complex numbers.
- The complex roots of quadratic equations with real coefficients occur in complex conjugate pairs. Hence, if is a complex root of a quadratic equation with real coefficients, we know that is also a root.
- Given a single complex root to a quadratic equation with real coefficients, we can reconstruct the original equation. In particular, if one of the roots is , one possible quadratic equation is