Problem 17 - Advanced Engineering Mathematics

Question

In Problems 17-20, write the given number in the form \(a+i b\). $$ i(2-3 i)^{2}(4+2 i) $$

Step-by-Step Solution

Answer

The expression simplifies to \(58 + 4i\).

Step 1: Simplify the Squared Term

First, simplify the expression \((2-3i)^2\). Use the formula for squaring a binomial: \((a-b)^2 = a^2 - 2ab + b^2\).Substitute \(a = 2\) and \(b = 3i\):\[ (2 - 3i)^2 = 2^2 - 2 \cdot 2 \cdot 3i + (3i)^2 \] \[ = 4 - 12i + 9i^2 \] Since \(i^2 = -1\), we have:\[ 9i^2 = 9(-1) = -9 \]So, the expression simplifies to: \[ 4 - 12i - 9 \] \[ = -5 - 12i \].

Step 2: Multiply with the Distributive Property

Now, use the distributive property to multiply \(i(-5 - 12i)(4 + 2i)\). Focus initially on \((-5 - 12i)(4 + 2i)\):\[(a+b)(c+d) = ac + ad + bc + bd\]Substituting \(a = -5\), \(b = -12i\), \(c = 4\), and \(d = 2i\):\[ (-5)(4) + (-5)(2i) + (-12i)(4) + (-12i)(2i)\]Which is:\[ -20 - 10i - 48i - 24i^2\]Calculate \(-24i^2\):\[ -24i^2 = -24(-1) = 24\], so:\[ -20 - 10i - 48i + 24 = 4 - 58i \].

Step 3: Multiply by i

Next, multiply the result by \(i\):\[ i(4 - 58i) = i \cdot 4 - i \cdot 58i \]Calculate it:\[ 4i - 58i^2\] Since \(i^2 = -1\), \(-58i^2 = 58 \):Thus, you get:\[ 58 + 4i \].

Step 4: Write the Final Result in Standard Form

The expression \(58 + 4i\) is already in the form of \(a + bi\), where \(a = 58\) and \(b = 4\). Thus, the expression is already in the required form.

Key Concepts

Binomial Expansion

The binomial expansion is an important algebraic process used to expand expressions raised to a power. It is especially useful in dealing with powers of sums. In mathematical terms, a binomial is an expression of the form \((a + b)\). When squared or raised to higher powers, each term in the binomial is expanded and simplified using the formula.
One common formula in binomial expansion is

• \((a - b)^2 = a^2 - 2ab + b^2\)

For example, when expanding \((2 - 3i)^2\), we identify:

• \(a = 2\)
• \(b = 3i\)

Applying the formula results in:

• \(2^2 - 2 \times 2 \times 3i + (3i)^2\)
• \(= 4 - 12i + 9i^2\)

Since \(i^2 = -1\), the expression simplifies further to \(4 - 12i - 9\), which equals \(-5 - 12i\).
Understanding binomial expansion is crucial when working with polynomials and expressions involving complex numbers.

Distributive Property

The distributive property is a basic principle in mathematics that allows you to multiply a single term across terms within a set of parentheses. In expressions involving complex numbers, such as \((-5 - 12i)(4 + 2i)\), this property helps in simplifying products of binomials.
The pattern to follow is:

• \((a+b)(c+d) = ac + ad + bc + bd\)

So, for \((-5 - 12i)(4 + 2i)\), we perform the following calculations:

• Multiply \(-5\) by \(4\) to get \(-20\)
• Multiply \(-5\) by \(2i\) to get \(-10i\)
• Multiply \(-12i\) by \(4\) to get \(-48i\)
• Multiply \(-12i\) by \(2i\) to get \(-24i^2\)
• Combine these results to get \(-20 - 10i - 48i - 24i^2\)

Remember \(i^2 = -1\), so \(-24i^2\) becomes \(24\). The expression simplifies to \(4 - 58i\).
This property is useful in algebra to simplify complex polynomials by breaking them down into simpler, more manageable pieces.

Imaginary Unit

The imaginary unit, denoted by \(i\), is a mathematical concept used to extend the real number system. The defining property of \(i\) is that \(i^2 = -1\). This means the square of an imaginary unit is a negative real number, which is impossible in the realm of real numbers.
In complex numbers, which take the form

• \(a + bi\)

, \(a\) is the real part and \(bi\) is the imaginary part, where \(b\) is a real number multiplier of the imaginary unit \(i\).
Let’s see an example with the exercise:

• When squaring a term like \(3i\), it becomes \(9i^2\)
• Using \(i^2 = -1\), it simplifies to \(-9\)

Similarly, when simplifying \(-24i^2\), you get \(-24(-1) = 24\), showcasing how \(i\) helps comprehend complex multiplication.
Understanding the imaginary unit is necessary for any study involving imaginary or complex numbers, including fields such as engineering, physics, and advanced mathematics.