For Exercises 11-38, use the square root property to solve...

Key Concepts

Square Root Property

The square root property is a method for solving equations where a variable is squared by isolating the squared term and then taking the square root of both sides of the equation, remembering to include both the positive and negative square roots. This property is particularly useful in solving quadratic equations that are easily reducible to the form x² = k.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, generally written in the form ax² + bx + c = 0. They can be solved using various methods such as factoring, completing the square, the quadratic formula, or the square root property. Understanding the structure of quadratic equations is fundamental in algebra.

Complex Number Format

Complex number format refers to expressing numbers in the form a ± bi, where a represents the real part and bi represents the imaginary part. This format is essential when dealing with quadratic equations that have a negative discriminant, resulting in non-real or complex solutions.

Transcript

00:01 We're being asked to show that one of the solutions to our given equation is root 2 over 2 plus root 2 over 2 times i.

00:07 Well, to prove that it would be a solution, we just need to substitute it into our equation for x and show that we would have a true statement.

00:14 That's how you can always check if something's a solution.

00:16 So if we were to do that, we would have to take the root 2 over 2 plus root 2 over 2, raise to the 8th power minus 1, and show that it would be equal to 0.

00:30 Well, what we need to do is we need to multiply this out.

00:34 So what i'm going to do is i'm going to rewrite this as root 2 over 2 plus root 2 over 2 times i, then times root 2 over 2 plus root 2 over 2 times i.

00:45 And that really would be root 2 plus root 2 over 2 times i squared.

00:51 So for these next ones, i'm going to rewrite it as that.

00:53 So i'm going to have root 2 over 2 plus root 2 over 2 times i squared, because that would be to the 4.

01:00 Root 2 over 2 plus root 2 over 2 times i squared again, and then another root 2 over 2 plus root 2 over 2 times i squared, because that would give us our eighth of them.

01:13 So now all i'm going to actually do is multiply out the first two, because its result will be the same for each of these other expressions.

01:21 So let's foil it.

01:22 Well, root 2 over 2 times root 2 over 2 would be 2 4ps, which is really just 1 1⁄.

01:28 So we're going to have 1 1 1�.

01:29 Then we have root 2 over 2 times root 2 over 2 i, which is going to be just 1 1 half i.

01:37 Then we'll have root 2 over 2 times root 2 over 2, which is also positive 1 1⁄2.

01:43 And lastly, root 2 over 2 times root 2 over 2 i will be just positive 1⁄2.

01:52 Then i'll bring down all the other terms, root 2 over 2 plus root 2 over 2 i squared.

01:58 Then we'll have root 2 over 2 plus root 2 over 2 i squared root 2 over 2 plus root 2 over 2 i squared minus 1 equals 0.

02:11 So now let's go ahead and combine late terms here...

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