Things I’ve figured out in math that I want to share so that you’ll have a better understanding or shortcuts to make life easier.

__Quadratic equation shortcut if c is 0__

Case: ax^2 + bx = 0

Answer: x = 0 or x = -b/a

__Imaginary Powers__

So the idea is to get to a power of 2 to power of x, and with an even exponent change to 1 and with an odd exponent, stay as -1

Remembering, i^0=1, i^1=i, i^2=-1, i^3=-i

i^623

i^620 * i^3

(i^2)^310 * i^3

Since 310 is even, it must be 1

1 * i^3

1 * -i

Answer: -i

or

i^622

(i^2)^311

Since 311 is odd, it must be -1

Answer: -1

__Negative Square Roots + Multiplication (Imaginary Numbers)__

I’m going to pose this question to you, what 2 numbers form a negative number? If you thought a negative and a positive number, you are correct. So, with that in mind, the only way to solve a negative square root tells us that there must be 2 solutions.

So solving this you get…

sqrt(-36) = i * sqrt(36) = 6i

Now, while this looks like it has one solution, keeping in mind the rule of a negative and positive, you’ll realize why this next bit isn’t allowed.

sqrt(-18) * sqrt(-2)

sqrt(-18 * -2)

sqrt(36)

Wrong: 6

You can’t multiply two solutions simultaneously which is why you have to break it down into single step processes

sqrt(-18) * sqrt(-2)

i * sqrt(18) * i * sqrt(2)

i^2 * sqrt(36)

i^2 * 6

Answer: -6