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## Algebra 2

### Course: Algebra 2>Unit 2

Lesson 6: Quadratic equations with complex solutions

# Complex numbers: FAQ

## What is the imaginary unit $i$i?

The imaginary unit i is a special number that, when squared, gives us minus, 1. It's a key part of complex numbers, which we'll learn about in this unit.

## What are complex numbers?

Complex numbers are numbers that have both a "real" part and an "imaginary" part. We can write them as a, plus, b, i, where a is the real part and b, i is the imaginary part.

## What are some real-world applications of complex numbers?

Complex numbers are used in a wide variety of fields, such as engineering, physics, and mathematics. They are especially useful for modeling certain types of electrical circuits and for analyzing signals in electronics.

## What is the complex plane?

The complex plane is a way of visualizing complex numbers. We plot the real part of the number on the horizontal axis, and the imaginary part on the vertical axis.

## How do we add and subtract complex numbers?

To add or subtract two complex numbers, we combine their real parts and their imaginary parts separately. For example, left parenthesis, 2, plus, 3, i, right parenthesis, plus, left parenthesis, 4, minus, 2, i, right parenthesis, equals, 6, plus, i.

## How do we multiply complex numbers?

To multiply two complex numbers, we use the distributive property to multiply all the terms together. For example:
\begin{aligned} (2 + 3i)(4 - 2i) &= 8 - 4i + 12i - 6i^2 \\\\ &= 8 - 4i + 12i - 6(-1) \\\\ &= 8 + 8i + 6 \\\\ &= 14 + 8i \end{aligned}

## Why do we need complex numbers to solve some quadratic equations?

Some quadratic equations don't have any "real" solutions, but they do have complex solutions. For example, x, squared, plus, 1, equals, 0 doesn't have any real solutions, but it does have the two complex solutions x, equals, plus minus, i.

## Want to join the conversation?

• that last section got me
• What does ∓ mean versus ±?
• When both symbols appear in the same expression, it indicates that when the addition sign of one symbol is used, the negative sign of the other should be used (and vice versa).

For example, 1 ± 2 ∓ 4 is equal to 1 + 2 - 4 and 1 - 2 + 4.
• I still don't get this
• me either
• I understand the Addition, Subtraction and Multiplication. I have a problem for the Solving Quadratic Equation: Complex Roots.It is more complicated and challenging than the others. I still need some help on that and I want to ask if there are any method I could use aside from this method? Like another way I could use to solve these solutions.
• Hi, @Dzidzorgbe Adonu 1, I see that you are having trouble with this concept. But this is similar to finding the roots of a quadratic equation using the methods: completing the square, using the quadratic formula, factoring, and using the square roots. Instead, the only complicated thing that is added to this concept is imaginary numbers, the unit i, and complex numbers. Like the article said, not all solutions to finding the roots/zeros of a quadratic equation going to be "real", so sometimes you'll have to include the imaginary unit i. To me, I find this difficult, but as you practice doing problems on this, you can guarantee that it'll be easier. To do this as long as you know how to solve quadratic equations using the different methods and knowing about radicals, which is all Algebra I, then this should be easy. Now obviously it won't be easy at the beginning, but it gives you an advantage for know the basics.
(1 vote)
• Can we divide imaginary numbers as well as multiply?
• Yes. Dividing is the same as multiplying that number's reciprocal.

For example, 10i / 5i = 2.

For example, 10i / 5 = 2i.

For example, 10 / 5i = 2 / i.
• For the imaginary unit "i" does it not matter how big the number is as long as I divide it by 4?
(1 vote)
• I'm assuming you're talking about the power of i. If so then you would be correct. To determine what the exponent equates to, look at the remainder after dividing the power of i by 4.
For example, say you want to find the value of i^87. 87 has a remainder of 3 when divided by 4. i^3 = -i. Therefore i^87 is equal to -i.
Hope this cleared things up!
• Are imaginary numbers considered irrational?
• No, the irrational numbers are all the real numbers that are not rational. As imaginary numbers are distinct from the set of real numbers (excluding 0), they cannot be irrational.
• I keep getting stuck on square rooting numbers that become complex numbers. the last one was square rooting
52
for some reason the number it square rooted to was
13
how the hell did that happen!! I didn't really know how to tackle that. How can i get the answer in the easiest was possible
(1 vote)
• 52=4·13, so √52=√(4·13)=√4·√13=2√13
Note this is unrelated to complex numbers, this is properties of square roots by themselves. I suggest reviewing those.