Problem 52 - Algebra A Combined Function

Question

Multiply. Write your answers in the form \(a+b i\). $$ (4+7 i)^{2} $$

Step-by-Step Solution

Answer

The expression is equal to \(-33 + 56i\).

Step 1: Understand the Formula

To multiply a complex number by itself, use the formula \((a + b i)^2 = a^2 + 2ab i + (b i)^2\). Here, \(a = 4\) and \(b = 7\).

Step 2: Square the Real Part

Using the formula, calculate \(a^2\):\[4^2 = 16\]

Step 3: Calculate the Product of Real and Imaginary Parts

Calculate \(2ab i\):\[2 \cdot 4 \cdot 7 i = 56i\]

Step 4: Square the Imaginary Part

Calculate \((b i)^2\):\((7 i)^2 = 49 i^2\). Recall that \(i^2 = -1\), thus \(49 i^2 = -49\).

Step 5: Combine the Results in the Complex Form

Combine the results from Steps 2, 3, and 4 into the complex number formal: \(16 + 56i - 49\ = -33 + 56i\).

Key Concepts

Multiplying Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part, usually written in the form \(a + bi\), where \(a\) is the real component and \(bi\) is the imaginary component. Multiplying complex numbers involves distributing the terms and using the property \(i^2 = -1\).
To multiply two complex numbers, use the distributive property, just like multiplying binomials. For instance, if you have \((a + bi)(c + di)\), you can expand this expression as:

• First, multiply \(a\) by \(c\), yielding \(ac\).
• Next, multiply \(a\) by \(di\), giving \(adi\).
• Then, multiply \(bi\) by \(c\), resulting in \(bci\).
• Finally, multiply \(bi\) by \(di\), producing \((bdi^2)\).

Because \(i^2 = -1\), simplifying the result will involve replacing \(i^2\) with \(-1\) to produce a combined real and imaginary part.

Complex Number Formula

The complex number formula, \((a + b i)^2 = a^2 + 2ab i + (b i)^2\), is used specifically when a complex number is multiplied by itself.
This formula is essential when expanding the square of a complex number, as it directly leads to the necessary calculations.Here’s how the formula works:

• The \(a^2\) part represents squaring the real part of the complex number.
• The \(2abi\) term results from combining the real and imaginary parts curtailed with a factor of 2.
• The term \((b i)^2\) reflects squaring the imaginary part; again it reminds you to use \(i^2 = -1\) to obtain the correct result.

By applying this formula, you can easily convert a complex number squared into its standard form \(a + bi \), elegantly and correctly.

Imaginary Unit

The imaginary unit is denoted by \(i\) and it is defined as the square root of -1. This definition might seem unusual because negative numbers do not have real square roots. However, in complex number mathematics, \(i\) allows for imaginary components.
One crucial rule with \(i\) is that \(i^2 = -1\). This property is essential for simplifying expressions involving complex numbers as seen in calculations.
The use of imaginary numbers, even if they sound strange, extends the scope of mathematics, particularly in fields like engineering, physics, and computer science, where they offer solutions that real numbers alone cannot address. Whenever you multiply and manage terms like \((bi)^2\), remember this pivotal property \(i^2 = -1\) for a successful solution.