In Exercises 9–20, find each product and write the result in s...

Question

In Exercises 9–20, find each product and write the result in standard form. (2 + 3i)²

Accepted Answer

Welcome back. Everyone in this problem. We want to square 15 plus two I and express our answer in the standard form for complex numbers. For our answer choices, A is 209 plus 120. IB is 224 plus 30. IC is 161 plus 24 I and D is 221 plus 60. I. Now, what do we know here? Well, we are squaring a complex number 15 plus two IR squared. And we want to write that complex number in standard form. Recall that the standard form of a complex number is when our complex number is written in the form A plus B I where A is the real part and B is the imaginary part. So we want to square our complex number and ensure that at the end of the day, it's in that form. So let's go ahead and do that. Now, look at our expression here. We, we are squaring a binomial expression. Doesn't this look similar to something you've done before? Doesn't it look like our perfect square? Because recall that if we're squaring an expression, let's say X plus Y all squared, then we can rewrite it as X squared plus two multiplied by XY plus Y squared. That's the perfect square. When we look at our expression, if we think of X as 15 and Y as two, I, then we can rewrite it as 15 squared plus two multiplied by 15, multiplied by two I plus two I all squared. If we take that a step further, 15 squared is 225 2, multiplied by 15 is 30 multiplied by two I, that's 60 I and two I all squared equals four I squared. No, we can take that even one step further because notice or recall rather again, a property of complex numbers that I squared equals negative one. Because if you can remember given, I let me write it to, just to jog your memory, given I equals the square of negative one, then I squared would be equal to the square of negative one or squared, which would have just been negative one. So we can apply that property because no, that means we can rewrite our expression as 225 plus 60 I plus four multiplied by negative one that becomes negative for and negative four is like 2 25. We can combine the consonants, 2 25 minus four equals 221. So our final answer is 221 plus 60 I which is in standard form. And that would have been answer choice D thanks a lot for watching everyone. I hope this video helped.

In Exercises 9–20, find each product and write the result in standard form. (2 + 3i)²

Key Concepts

Complex Numbers and Standard Form

Binomial Expansion (Square of a Binomial)

Properties of the Imaginary Unit i

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