Problem 60 - Algebra 2

Question

Simplify. $$ \frac{(10+i)^{2}}{4-i} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(22 + \frac{2}{17} + 10i + \frac{9i}{17}\).

Step 1: Expand the Numerator

We start by expanding \((10+i)^2\). This is a binomial expansion:\[(10+i)^2 = 10^2 + 2(10)(i) + i^2\]Using the fact that \(i^2 = -1\), the expansion becomes:\[100 + 20i + (-1) = 99 + 20i\].

Step 2: Rationalize the Denominator

To simplify \(\frac{99+20i}{4-i}\), we need to rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of \((4-i)\) is \((4+i)\). Multiply both the numerator and the denominator:\[\frac{(99+20i)(4+i)}{(4-i)(4+i)}\].

Step 3: Multiply the Denominator

Multiply \((4-i)(4+i)\) using the difference of squares formula:\[(4-i)(4+i) = 4^2 - i^2 = 16 + 1 = 17\].

Step 4: Multiply the Numerator

Expand \((99+20i)(4+i)\) using the distributive property:\[(99 \cdot 4) + (99 \cdot i) + (20i \cdot 4) + (20i \cdot i)\]Calculate each part:\[396 + 99i + 80i + 20(-1)\]Combine and simplify:\[396 + 179i - 20 = 376 + 179i\].

Step 5: Write the Simplified Expression

Now, write the simplified expression:\[\frac{376+179i}{17}\]Separate into real and imaginary parts:\[\frac{376}{17} + \frac{179i}{17}\]Perform division:\[22.1176 + 10.5294i\]However, leave it in fractional form:\[22 + \frac{2}{17} + 10i + \frac{9i}{17}\].

Key Concepts

Rationalize the Denominator

When dealing with complex numbers, having a real number as the denominator makes it simpler to handle and interpret. This process is called rationalizing the denominator. To do this, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number like \(a + bi\) is \(a - bi\). By multiplying, it removes the imaginary part from the denominator.
For example, in the fraction \(\frac{99+20i}{4-i}\), the conjugate of \(4-i\) is \(4+i\). By multiplying the numerator and the denominator by \(4+i\), the imaginary unit \(i\) in the denominator cancels out, leaving you with a real number, thus simplifying the expression.

Binomial Expansion

Binomial expansion refers to expanding expressions that have a binomial raised to a power, such as \((a+b)^2\). This is particularly useful for expressions like \((10+i)^2\). The formula follows:

• \( (a+b)^2 = a^2 + 2ab + b^2 \)

In the example \((10+i)^2\), we plug in 10 for \(a\) and \(i\) for \(b\), resulting in \(10^2 + 2(10)(i) + i^2\). Knowing that \(i^2 = -1\) helps convert complex numbers into a simplified form: \(100 + 20i - 1\). Overall, this process helps break down complex operations in a systematic and understandable way.

Difference of Squares

The difference of squares is a unique formula that simplifies multiplication involving complex conjugates. It is written as:

• \((a-b)(a+b) = a^2 - b^2\)

When you have complex numbers, such as \((4-i)(4+i)\), the formula helps by eliminating the middle terms and just subtracting the square of \(i\), that is \(-i^2 = 1\). Hence, you calculate this as \(4^2 - (-i^2)\), which simplifies to \(16 + 1 = 17\). By this method, the expression can be reduced efficiently, transforming a complex denominator into a real number.

Distributive Property

The distributive property is a fundamental algebraic approach that lets us expand expressions. It's particularly useful in complex arithmetic to simplify multiplications like \((99+20i)(4+i)\). This property states:

• \(a(b + c) = ab + ac\)

Thus, for \((99+20i)(4+i)\), you distribute each term in the first binomial to every term in the second binomial:

• \((99)(4) + (99)(i) + (20i)(4) + (20i)(i)\)

This leads to individual calculations: \(396 + 99i + 80i + 20\cdot(-1)\), which simplify to \(376 + 179i\). By breaking down the multiplication into smaller, manageable parts, complex numbers become easier to handle.