Problem 35 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ (-2+3 i)^{2} $$

Step-by-Step Solution

Answer

The expression (-2 + 3i)^{2} in standard form is -5 - 12i.

Step 1: Understand the Expression

The expression given is (-2 + 3i)^{2}, which is a complex number raised to the power of 2. Our task is to expand this expression and write it in standard form, which means in the form a + bi, where a and b are real numbers.

Step 2: Apply the Binomial Theorem

The expression (-2 + 3i)^{2} can be expanded using the binomial theorem. According to the formula for squaring a binomial, (a + b)^{2} = a^{2} + 2ab + b^{2}. Here, a = -2 and b = 3i. Substitute these into the formula.

Step 3: Calculate the Squares

Calculate each square in the expansion: (-2)^{2} = 4 (3i)^{2} = 9i^{2} Remembering that i^{2} = -1, the value of 9i^{2} becomes 9(-1) or -9.

Step 4: Calculate the Middle Term

Calculate the middle term, 2ab = 2(-2)(3i) = -12i.

Step 5: Combine Terms

Combine all the terms from the expansion: 4 (from a^{2}) + (-12i) (from 2ab) - 9 (from b^{2}) Simplify to: (4 - 9) + (-12i) = -5 - 12i.

Step 6: Write in Standard Form

The expression is now simplified and in standard form: -5 - 12i. In this form, a = -5 and b = -12.

Key Concepts

Standard Form

Complex numbers are written in standard form as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. This form helps in easily understanding and performing operations on complex numbers.

• The real part, \(a\), is the component that doesn't involve \(i\).
• The imaginary part, \(bi\), involves the imaginary unit \(i\).
• Standard form combines both these parts to represent the complex number clearly.

In the exercise \((-2 + 3i)^2\), after following the calculations we eventually reach the standard form \(-5 - 12i\). Here, \(a = -5\) (the real part) and \(b = -12\) (the coefficient of \(i\)). Thus, the expression is successfully simplified to reflect both real and imaginary parts in standard form.

Binomial Theorem

The binomial theorem helps expand expressions that are raised to a power, especially when they involve two terms. In mathematics, it states:
\( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \).
For squaring a binomial, a simplified formula is used:

• The formula for squaring \((a + b)^2\) is \(a^2 + 2ab + b^2\).
• This means you must calculate each term: the square of the first term \(a^2\), double the product of both terms \(2ab\), and the square of the second term \(b^2\).

Applying this to our complex numbers, \((-2 + 3i)^2\):

• First term: \((-2)^2 = 4\)
• Middle term: \(2 \times (-2) \times (3i) = -12i\)
• Last term: \((3i)^2 = 9i^2 = -9\)

Putting it all together results in: \(4 - 9 + (-12i) = -5 - 12i\), illustrating the effective use of the binomial theorem in expanding the expression.

Imaginary Unit

The imaginary unit \(i\) is fundamental in dealing with complex numbers. It is defined by the property \(i^2 = -1\). This property is used extensively when performing arithmetic with complex numbers.

• In any complex number, the term with \(i\) gives it its 'imaginary' characteristic.
• When it comes to powers, especially squares, remember \(i^2 = -1\). This helps convert products involving \(i\) back into real numbers.

In our specific example, the step \((3i)^2\) results as:

• \((3i)^2 = 3^2 \times i^2 = 9 \times (-1) = -9\).

By substituting \(i^2 = -1\), the expression not only simplifies but also integrates the imaginary part back into a real form, allowing us to write it conveniently in standard form \(a + bi\). This concept is crucial when mastering calculations involving complex numbers.