Miss Harback writes the equation \begin{align*}5x^2 + 125 = 0\end{align*} on the board. She asks the class how many solutions the equation has and what type they are.

Corrine says the equation has two real solutions. Drushel says the equation has a double root, so only one solution. Farrah says the equation has two imaginary solutions.

Which one of them is correct?

### Quadratic Equations with Complex Number Solutions

When you solve a quadratic equation, there will always be two answers. Until now, we thought the answers were always real numbers. In actuality, there are quadratic equations that have imaginary solutions as well. The possible solutions for a quadratic are:

2 real solutions

\begin{align*}x^2-4 &= 0\\ x &= -2,2\end{align*}

Double root

\begin{align*}x^2+4x+4 &= 0\\ x &= -2,-2\end{align*}

2 imaginary solutions

\begin{align*}x^2+4 &= 0\\ x &= -2i,2i\end{align*}

Let's solve the following quadratic equations.

- Solve \begin{align*}3x^2+27=0\end{align*}.

First, factor out the GCF.

\begin{align*}3(x^2+9)=0\end{align*}

Now, try to factor \begin{align*}x^2 + 9\end{align*}. Rewrite the quadratic as \begin{align*}x^2 + 0x + 9\end{align*} to help. There are no factors of 9 that add up to 0. Therefore, this is not a factorable quadratic. Let’s solve it using square roots.

\begin{align*}3x^2+27 &= 0\\ 3x^2 &= -27\\ x^2 &= -9\\ x &= \pm \sqrt{-9}= \pm 3i\end{align*}

**Quadratic equations with imaginary solutions are never factorable.**

- Solve \begin{align*}(x-8)^2=-25\end{align*}.

Solve using square roots.

\begin{align*}(x-8)^2 &= -25\\ x-8 &= \pm 5i\\ x &= 8 \pm 5i\end{align*}

- Solve \begin{align*}2(3x-5)+10=-30\end{align*}.

Solve using square roots.

\begin{align*}2(3x-5)^2+10 &=-30\\ 2(3x-5)^2 &=-40\\ (3x-5)^2 &=-20\\ 3x-5 &=\pm2i\sqrt{5}\\ 3x &=5\pm2i\sqrt{5}\\ x &=\frac{5}{3}\pm\frac{2\sqrt{5}}{3}i\end{align*}

### Examples

#### Example 1

Earlier, you were asked to find which student is correct.

To solve \begin{align*}5x^2+125=0\end{align*}, we first need to factor out the GCF.

\begin{align*}5(x^2+25)=0\end{align*}

Now, try to factor \begin{align*}x^2 + 25\end{align*}. Rewrite the quadratic as \begin{align*}x^2 + 0x + 25\end{align*} to help. There are no factors of 25 that add up to 0. Therefore, this is not a factorable quadratic. Let’s solve it using square roots.

\begin{align*}5x^2 + 125 &= 0\\ 5x^2 &= -125\\ x^2 &= -25\\ x &= \pm \sqrt{-5}= \pm 5i\end{align*}

The equation has two roots and both of them are imaginary, so Farrah is correct.

#### Example 2

Solve \begin{align*}4(x-5)^2+49=0\end{align*}.

\begin{align*}4(x-5)^2+49 &=0\\
4(x-5)^2 &=-49\\
(x-5)^2 &=-\frac{49}{4}\\
x-5 &=\pm\frac{7}{2}i\\
x &=5\pm\frac{7}{2}i\end{align*}

#### Example 3

Solve \begin{align*}-\frac{1}{2}(3x+8)^2-16=2\end{align*}.

\begin{align*}-\frac{1}{2}(3x+8)^2-16 &=2\\
-\frac{1}{2}(3x+8)^2 &=18\\
(3x+8)^2 &=-36\\
3x+8 &=\pm6i\\
3x &=-8\pm6i\\
x &=-\frac{8}{3}\pm2i\end{align*}

Both of the quadratic equations in Examples 2 and 3 can be solved by using square roots.

### Review

Solve the following quadratic equations.

- \begin{align*}x^2=-9\end{align*}
- \begin{align*}x^2+8=3\end{align*}
- \begin{align*}(x+1)^2=-121\end{align*}
- \begin{align*}5x^2+16=-29\end{align*}
- \begin{align*}14-4x^2=38\end{align*}
- \begin{align*}(x-9)^2-2=-82\end{align*}
- \begin{align*}-3(x+6)^2+1=37\end{align*}
- \begin{align*}4(x-5)^2-3=-59\end{align*}
- \begin{align*}(2x-1)^2+5=-23\end{align*}
- \begin{align*}-(6x+5)^2=72\end{align*}
- \begin{align*}7(4x-3)^2-15=-68\end{align*}
- If a quadratic equation has \begin{align*}4 - i\end{align*} as a solution, what must the other solution be?
- If a quadratic equation has \begin{align*}6 + 2i\end{align*} as a solution, what must the other solution be?
**Challenge**Recall that the factor of a quadratic equation has the form \begin{align*}(x\pm m)\end{align*} where \begin{align*}m\end{align*} is any number. Find a quadratic equation that has the solution \begin{align*}3 + 2i\end{align*}.- Find a quadratic equation that has the solution \begin{align*}1-i\end{align*}.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 5.10.