# 5.3.3: Quadratic Formula and Complex Sums

• • Contributed by CK12
• CK12

Solve quadratic equations with complex roots, and add and subtract complex numbers.

## Complex Roots of Quadratic Functions

The quadratic function $$y=x^2−2x+3$$ (shown below) does not intersect the x-axis and therefore has no real roots. What are the complex roots of the function? Figure $$\PageIndex{1}$$

### Complex Roots of Quadratic Functions

Recall that the imaginary number, $$i$$, is a number whose square is –1:

$$\textcolor{red}{i^2=−1}$$ and $$\textcolor{red}{i=\sqrt{−1}}$$

The sum of a real number and an imaginary number is called a complex number. Examples of complex numbers are $$5+4i$$ and $$3−2i$$. All complex numbers can be written in the form a+bi where a and b are real numbers. Two important points:

• The set of real numbers is a subset of the set of complex numbers where $$b=0$$. Examples of real numbers are 2,7,12,−4.2.
• The set of imaginary numbers is a subset of the set of complex numbers where a=0. Examples of imaginary numbers are $$i$$, $$−4i$$, $$\sqrt{2}i$$.

This means that the set of complex numbers includes real numbers, imaginary numbers, and combinations of real and imaginary numbers.

When a quadratic function does not intersect the x-axis, it has complex roots. When solving for the roots of a function algebraically using the quadratic formula, you will end up with a negative under the square root symbol. With your knowledge of complex numbers, you can still state the complex roots of a function just like you would state the real roots of a function.

Let's solve the quadratic equation: $$m^2−2m+5=0$$

You can use the quadratic formula to solve. For this quadratic equation, $$a=1$$, $$b=−2$$, $$c=5$$.

$$\begin{array}{l} m=\dfrac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \\ m=\dfrac{-(\textcolor{red}{-2}) \pm \sqrt{(\textcolor{red}{-2})^{2}-4(\textcolor{red}{1})(\textcolor{red}{5})}}{2(\textcolor{red}{1})} \\ m=\dfrac{2 \pm \sqrt{4-20}}{2} \\ m=\dfrac{2 \pm \sqrt{-16}}{2} \quad \sqrt{-16}=\sqrt{16} \times i=4 i \\ m=\dfrac{2 \pm 4 i}{2} \\ m=1 \pm 2 i \\ m=1+2 i \text { or } m=1-2 i \end{array}$$

There are no real solutions to the equation. The solutions to the quadratic equation are $$1+2i$$ and $$1−2i$$.

#### Now, let's solve the following equation by rewriting it as a quadratic and using the quadratic formula:

$$\dfrac{3}{e+3}−\dfrac{2}{e+2}=1$$

To rewrite as a quadratic equation, multiply each term by $$(e+3)(e+2)$$.

\begin{aligned} \dfrac{3}{e+3} \textcolor{red}{(e+3)(e+2)}−2e+2\textcolor{red}{(e+3)(e+2)}=1\textcolor{red}{(e+3)(e+2)} \\ 3(e+2)−2(e+3)=(e+3)(e+2) \end{aligned}

Expand and simplify.

\begin{aligned} 3e+6−2e−6=e^2+2e+3e+6 \\ e^2+4e+6=0 \end{aligned}

$$\begin{array}{l} e=\dfrac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \\ e=\dfrac{-(\textcolor{red}{4}) \pm \sqrt{(\textcolor{red}{4})^{2}-4(\textcolor{red}{1})(\textcolor{red}{6})}}{2(\textcolor{red}{1})} \\ e=\dfrac{-4 \pm \sqrt{16-24}}{2} \\ e=\dfrac{-4 \pm \sqrt{-8}}{2} \quad \sqrt{-8}=\sqrt{8} \times i=\sqrt{4 \cdot 2} \times i=2 i \sqrt{2} \\ e=\dfrac{-4 \pm 2 i \sqrt{2}}{2} \\ e=-2 \pm i \sqrt{2} \\ e=-2+i \sqrt{2} \text { or } e=-2-i \sqrt{2} \end{array}$$

There are no real solutions to the equation. The solutions to the equation are −2+i\sqrt{2} and −2−i\sqrt{2}

#### Finally, let's sketch the graph of the following quadratic function. What are the roots of this function?

$$y=x^2−4x+5$$

Use your calculator or a table to make a sketch of the function. You should get the following: Figure $$\PageIndex{2}$$

As you can see, the quadratic function has no x-intercepts; therefore, the function has no real roots. To find the roots (which will be complex), you must use the quadratic formula.

For this quadratic function, $$a=1$$, $$b=−4$$, $$c=5$$.

$$\begin{array}{l} x=\dfrac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \\ x=\dfrac{-(\textcolor{red}{-4}) \pm \sqrt{(\textcolor{red}{-4})^{2}-4\textcolor{red}{1})(\textcolor{red}{5})}}{2(\textcolor{red}{1})} \\ x=\dfrac{4 \pm \sqrt{16-20}}{2} \\ x=\dfrac{4 \pm \sqrt{-4}}{2} \quad \sqrt{-4}=\sqrt{4} \times i=2 i \\ x=\dfrac{4 \pm 2 i}{2} \\ x=2 \pm i \\ x=2+i \text { or } x=2-i \end{array}$$

The complex roots of the quadratic function are 2+i and 2−i.

Example $$\PageIndex{1}$$

Earlier, you were asked to find the complex roots of $$y=x^2−2x+3$$.

Solution

To find the complex roots of the function $$y=x^2−2x+3$$, you must use the quadratic formula.

For this quadratic function, $$a=1$$, $$b=−2$$, $$c=3$$.

$$\begin{array}{l} x=\dfrac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \\ x=\dfrac{-(\textcolor{red}{-2}) \pm \sqrt{(\textcolor{red}{-2})^{2}-4(\textcolor{red}{1})(\textcolor{red}{3})}}{2(\textcolor{red}{1})} \\ x=\dfrac{2 \pm \sqrt{4-12}}{2} \\ x=\dfrac{2 \pm \sqrt{-8}}{2} \quad \sqrt{-8}=\sqrt{8} \times i=2 \sqrt{2} i \\ x=\dfrac{2 \pm 2 \sqrt{2} i}{2} \\ x=1 \pm \sqrt{2} i \end{array}$$

Example $$\PageIndex{2}$$

Solve the following quadratic equation. Express all solutions in simplest radical form.

$$2n^2+n=−4$$

Solution

$$2n^2+n=−4$$

Set the equation equal to zero.

$$2n^2+n+4=0$$

$$\begin{array}{l} x=\dfrac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \\ n=\dfrac{-(\textcolor{red}{1}) \pm \sqrt{(\textcolor{red}{1})^{2}-4(\textcolor{red}{2})(\textcolor{red}{4})}}{2(\textcolor{red}{2})} \\ n=\dfrac{-1 \pm \sqrt{1-32}}{4} \\ n=\dfrac{-1 \pm \sqrt{-31}}{4} \\ n=\dfrac{-1 \pm i \sqrt{31}}{4} \end{array}$$

Example $$\PageIndex{3}$$

Solve the following quadratic equation. Express all solutions in simplest radical form.

$$m^2+(m+1)^2+(m+2)^2=−1$$

Solution

$$m^2+(m+1)^2+(m+2)^2=−1$$

Expand and simplify.

\begin{aligned} m^2+(m+1)(m+1)+(m+2)(m+2)&=−1 \\ m^2+m^2+m+m+1+m^2+2m+2m+4&=−1 \\ 3m^2+6m+5&=−1 \end{aligned}

Write the equation in general form.

$$3m^2+6m+6=0$$

Divide by 3 to simplify the equation.

\begin{aligned} \dfrac{3m^2}{\textcolor{red}{3}}+\dfrac{6m}{\textcolor{red}{3}}+\dfrac{6}{\textcolor{red}{3}}&=\dfrac{0}{\textcolor{red}{3}} \\ m^2+2m+2&=0 \end{aligned}

$$\begin{array}{l} m=\dfrac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \\ m=\dfrac{-(\textcolor{red}{2}) \pm \sqrt{(\textcolor{red}{2})^{2}-4(\textcolor{red}{1})(\textcolor{red}{2})}}{2(\textcolor{red}{1})} \\ m=\dfrac{-2 \pm \sqrt{4-8}}{2} \\ m=\dfrac{-2 \pm \sqrt{-4}}{2} \\ m=\dfrac{-2 \pm 2 i}{2} \\ m=-1 \pm i \end{array}$$

Example $$\PageIndex{4}$$

Is it possible for a quadratic function to have exactly one complex root?

Solution

No, even in higher degree polynomials, complex roots will always come in pairs. Consider when you use the quadratic formula-- if you have a negative under the square root symbol, both the + version and the - version of the two answers will end up being complex.

## Review

1. If a quadratic function has 2 x-intercepts, how many complex roots does it have? Explain.
2. If a quadratic function has no x-intercepts, how many complex roots does it have? Explain.
3. If a quadratic function has 1 x-intercept, how many complex roots does it have? Explain.
4. If you want to know whether a function has complex roots, which part of the quadratic formula is it important to focus on?
5. You solve a quadratic equation and get 2 complex solutions. How can you check your solutions?
6. In general, you can attempt to solve a quadratic equation by graphing, factoring, completing the square, or using the quadratic formula. If a quadratic equation has complex solutions, what methods do you have for solving the equation?

Solve the following quadratic equations. Express all solutions in simplest radical form.

1. $$x^2+x+1=0$$
2. $$5y^2−8y=−6$$
3. $$2m^2−12m+19=0$$
4. $$−3x^2−2x=2$$
5. $$2x^2+4x=−11$$
6. $$−x^2+x−23=0$$
7. $$−3x^2+2x=14$$
8. $$x^2+5=−x$$
9. $$\dfrac{1}{2}d^2+4d=−12$$

To see the Review answers, open this PDF file and look for section 9.7.

## Vocabulary

Term Definition
complex number A complex number is the sum of a real number and an imaginary number, written in the form $$a+bi$$.
complex root A complex root is a complex number that, when used as an input ($$x$$) value of a function, results in an output ($$y$$) value of zero.
Imaginary Numbers An imaginary number is a number that can be written as the product of a real number and i.
Quadratic Formula The quadratic formula states that for any quadratic equation in the form $$ax^2+bx+c=0$$, $$x=\dfrac{−b\pm \sqrt{b^2−4ac}}{2a}$$.
Real Number A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.