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# Quadratic Formula and Complex Sums

## Conversion between a + bi and (a, b), (r, theta), and rcistheta.

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The quadratic function \begin{align*}y=x^2-2x+3\end{align*} (shown below) does not intersect the x-axis and therefore has no real roots. What are the complex roots of the function?

### Guidance

Recall that the imaginary number, \begin{align*}i\end{align*}, is a number whose square is –1:

\begin{align*}{\color{red}i^2 = -1}\end{align*} and \begin{align*}{\color{red}i=\sqrt{-1}}\end{align*}

The sum of a real number and an imaginary number is called a complex number. Examples of complex numbers are \begin{align*}5+4i\end{align*} and \begin{align*}3-2i\end{align*}. All complex numbers can be written in the form \begin{align*}a+bi\end{align*} where \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are real numbers. Two important points:

• The set of real numbers is a subset of the set of complex numbers where \begin{align*}b=0\end{align*}. Examples of real numbers are \begin{align*}2, 7, \frac{1}{2}, -4.2\end{align*}.
• The set of imaginary numbers is a subset of the set of complex numbers where \begin{align*}a=0\end{align*}. Examples of imaginary numbers are \begin{align*}i, -4i, \sqrt{2}i\end{align*}.

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.

#### Example A

Solve the following quadratic equation for \begin{align*}x\end{align*}.

Solution: You can use the quadratic formula to solve. For this quadratic equation, \begin{align*}a=1, b=-2, c=5\end{align*}.

There are no real solutions to the equation. The solutions to the quadratic equation are \begin{align*}1+2i \ and \ 1-2i\end{align*}.

#### Example B

Solve the following equation by rewriting it as a quadratic and using the quadratic formula.

\begin{align*} \frac{3}{e+3} - \frac{2}{e+2} =1\end{align*}

Solution: To rewrite as a quadratic equation, multiply each term by \begin{align*}(e + 3) (e + 2)\end{align*}.

Expand and simplify.

Solve using the quadratic formula. For this quadratic equation, \begin{align*}a=1, b=4, c=6\end{align*}.

There are no real solutions to the equation. The solutions to the equation are \begin{align*}-2+i\sqrt{2} \ and \ -2-i\sqrt{2}\end{align*}

#### Example C

Sketch the graph of the following quadratic function. What are the roots of the function?

Solution: Use your calculator or a table to make a sketch of the function. You should get the following:

As you can see, the quadratic function has no \begin{align*}x\end{align*}-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, \begin{align*}a=1, b=-4, c=5\end{align*}.

The complex roots of the quadratic function are \begin{align*}2+i \ and \ 2-i\end{align*}.

#### Concept Problem Revisited

To find the complex roots of the function \begin{align*}y=x^2-2x+3\end{align*}, you must use the quadratic formula.

For this quadratic function, \begin{align*}a=1, b=-2, c=3\end{align*}.

### Vocabulary

Complex Number
A complex number is a number in the form \begin{align*}a + bi\end{align*} where \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are real numbers and \begin{align*}i^2=-1\end{align*}.
Imaginary Number
An imaginary number is a number such that its square is a negative number.
\begin{align*}\sqrt{-16}\end{align*} is an imaginary number beacause its square is –16.
\begin{align*}\sqrt{-16}=i\sqrt{16} = 4i\end{align*}

### Guided Practice

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

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

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

1.

Set the equation equal to zero.

2.

Expand and simplify.

Write the equation in general form.

Divide by 3 to simplify the equation.

3. 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.

### Practice

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. \begin{align*}x^2+x+1=0\end{align*}
2. \begin{align*}5y^2-8y=-6\end{align*}
3. \begin{align*}2m^2-12m+19=0\end{align*}
4. \begin{align*}-3x^2-2x=2\end{align*}
5. \begin{align*}2x^2+4x=-11\end{align*}
6. \begin{align*}-x^2+x-23=0\end{align*}
7. \begin{align*}-3x^2+2x=14\end{align*}
8. \begin{align*}x^2+5=-x\end{align*}
9. \begin{align*}\frac{1}{2}d^2+4d=-12\end{align*}

### Vocabulary Language: English

complex number

complex number

A complex number is the sum of a real number and an imaginary number, written in the form $a + bi$.
complex root

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

Imaginary Numbers

An imaginary number is a number that can be written as the product of a real number and $i$.

The quadratic formula states that for any quadratic equation in the form $ax^2+bx+c=0$, $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$.
Real Number

Real Number

A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.