Problem 47 - College Algebra Essentials

Question

In Exercises \(45-50,\) perform the indicated operation(s) and write the result in standard form. $$ (2+i)^{2}-(3-i)^{2} $$

Step-by-Step Solution

Answer

-5 + 10i

Step 1: Squaring the first complex number

We start by squaring the first complex number \( (2 + i) \). This is done by multiplying every part of the complex number by every other part i.e. \( (2+i)*(2+i) \). This results in \( 4 + 2i + 2i + i^{2} \). Since \( i^{2} = -1 \), we simplify further to \( 4 + 4i - 1 = 3 + 4i \).

Step 2: Squaring the second complex number

Next, we square the second complex number \( (3 - i) \). This is done in the same way by multiplying every part of the complex number by every other part i.e. \( (3 -i)*(3 -i) \). This results to \( 9 - 3i - 3i + i^{2} \). Again, since \( i^{2} = -1 \), the expression simplifies to \( 9 - 6i - 1 = 8 - 6i \).

Step 3: Subtract the two results

Finally, subtract the result of squaring the second complex number \( 8 - 6i \) from the first result \( 3 + 4i \). This yields \( (3 + 4i) - (8 - 6i) = (3-8) + (4 + 6)i = -5 + 10i \).

Step 4: Final output in standard form

So, the result in its standard form is \( -5 + 10i \).

Key Concepts

Simplifying Complex Numbers

When working with complex numbers, simplifying them makes calculations easier and helps in understanding their properties. A complex number consists of a real part and an imaginary part, typically expressed as \(a + bi\). Simplifying complex numbers involves combining like terms and using the fact that \(i^2 = -1\).

For instance, if we have \( (2 + i)^2 \), we start by expanding the expression: \(2^2 + 2 \cdot 2 \cdot i + i^2\). This simplifies to \(4 + 4i - 1\) since \(i^2 = -1\), which further simplifies to \(3 + 4i\). You'll often need to remember to combine like terms and use the imaginary unit property to simplify complex numbers.

Standard Form of Complex Numbers

The standard form for complex numbers is \(a + bi\) where \(a\) is the real part and \(b\) is the imaginary part. It's essential to express complex numbers in this form for clarity and to follow mathematical conventions.

When we complete operations such as squaring complex numbers, we need to ensure the result is presented in the standard form. For instance, after squaring the complex numbers \(2 + i\) and \(3 - i\) and then subtracting the latter from the former, we arrive at \( -5 + 10i \). This result is in the standard form, demonstrating a real part, \( -5 \), and an imaginary part, \(10i\).

Squaring Complex Numbers

Squaring complex numbers involves multiplying the complex number by itself. This can be done using the FOIL method (First, Outside, Inside, Last), where you multiply each part of the first complex number with each part of the second.

Let's look at \( (3 - i)^2 \). We multiply \(3\) by \(3\), \(3\) by \( -i\), \( -i\) by \(3\), and \( -i\) by \( -i\), finally adding them all together. This gives us \(9 - 3i - 3i + i^2\), which simplifies to \(8 - 6i\) after recognizing that \(i^2 = -1\). Such methods are pivotal in managing complex expressions and should be practiced to gain proficiency.

Algebraic Operations with Complex Numbers

Complex numbers follow the same algebraic rules as real numbers. When performing operations like addition, subtraction, multiplication, and division, treat the real and imaginary parts separately.

In the exercise \( (2+i)^{2}-(3-i)^{2} \), we first squared both complex numbers separately. We then subtracted the second result from the first, which is \(3 + 4i\) minus \(8 - 6i\). By treating the real parts and the imaginary parts separately, we get \( (3-8) + (4 + 6)i = -5 + 10i \). This process is crucial for solving complex number operations correctly and with ease.