Rationalizing Complex Fractions: Conjugate Method (Example: (7+4i)/(2-3i))

Key idea: When you have a denominator that is a binomial with an imaginary part, multiply the fraction by the conjugate of the denominator to eliminate the imaginary part in the denominator.
Conjugate concept: The conjugate of a binomial a + bi is a − bi; for a − 3i the conjugate is a + 3i. In this example, the denominator is (2 − 3i) and its conjugate is (2 + 3i).
Why this works: (a − bi)(a + bi) = a^2 + b^2, which is a real number. This turns the denominator into a real constant and clears the imaginary part.
Overall plan:
- Multiply numerator and denominator by the conjugate of the denominator.
- Use the box/FOIL method to compute the products.
- Use i^2 = −1 to simplify the numerator and denominator.
- Convert the result to a + bi form by dividing the real and imaginary parts by the real denominator.

Given expression to simplify:
- Numerator:
- Denominator:
- In the transcript, the setup is:
- Numerator factor: (7 + 4i)
- Denominator factor: (2 − 3i)
- Conjugate to multiply by: (2 + 3i)
Step 1: Multiply by the conjugate of the denominator
- Expression becomes:
- $\frac{(7+4i)(2+3i)}{(2-3i)(2+3i)}$
Step 2: Compute the numerator product using the box/FOIL method
- Rectangle/box terms:
- 7 * 2 = 14
- 2 * 4i = 8i
- 7 * 3i = 21i
- 4i * 3i = 12i^2
- Combine: numerator = $14 + 8i + 21i + 12i^2 = 14 + 29i + 12i^2$
- Substitute i^2 = −1: $14 + 29i + 12(-1) = 14 + 29i - 12 = 2 + 29i$
Step 3: Compute the denominator product
- Rectangle/box terms:
- 2 * 2 = 4
- 2 * (3i) = 6i
- (−3i) * 2 = −6i
- (−3i) * (3i) = −9i^2
- Combine: denominator = $4 + 6i - 6i - 9i^2 = 4 - 9i^2$
- Substitute i^2 = −1: $4 - 9(-1) = 4 + 9 = 13$
Step 4: Form the new fraction and simplify
- Fraction after multiplying: $\frac{2 + 29i}{13}$
- Since the denominator is real, separate into a + bi form:
- $\frac{2}{13} + \frac{29}{13} i$
Step 5: Final answer
- The final simplified form is: $\frac{2}{13} + \frac{29}{13} i$

In the denominator, the cross terms cancel: $6i + (-6i) = 0$ , leaving a real number.
This is a direct consequence of using the conjugate: the product becomes a^2 + b^2 (a real value).
For this example, a = 2 and b = 3, so the real denominator is $a^2 + b^2 = 2^2 + 3^2 = 4 + 9 = 13$ .

Box (rectangular) method is a practical way to organize multiplications when you’re not sure about special products.
Special product to remember: (a − bi)(a + bi) = a^2 + b^2 (a real number).
If you don’t recognize the special product, you can still do the box/FOIL method and then simplify.

If you need to compute x + yi divided by a + bi:
- Multiply numerator and denominator by the conjugate a − bi:
- $\frac{x + yi}{a + bi} = \frac{(x + yi)(a - bi)}{(a + bi)(a - bi)} = \frac{(x + yi)(a - bi)}{a^2 + b^2}$
- The numerator expands to: (xa + yb) + i(ya - xb)
- Therefore: $\frac{x + yi}{a + bi} = \frac{xa + yb}{a^2 + b^2} + i \frac{ya - xb}{a^2 + b^2}$
If you need to divide by a − bi:
- Multiply numerator and denominator by the conjugate a + bi, yielding:
- Real part: (xa − yb)/(a^2 + b^2)
- Imag part: (xb + ya)/(a^2 + b^2)
In both cases, the denominator after multiplication becomes the real quantity $a^2 + b^2$ .

Always multiply both numerator and denominator by the same conjugate of the denominator.
Remember i^2 = −1 when simplifying the numerator.
Watch the signs carefully in the cross terms; in this example, the cross terms in the denominator cancel to give a real number.
After obtaining a complex numerator over a real denominator, you can split into real and imaginary parts by dividing each part by the real denominator.

Given a fraction of the form $(\dfrac{p + qi}{a - bi})$
1) Multiply top and bottom by the conjugate $(a + bi)$ .
2) Use FOIL/box to compute the products.
3) Replace i^2 with −1 and simplify.
4) Write the result in the form (\alpha + \beta i) by dividing by the real denominator (a^2 + b^2).
Apply this to the numbers in the example to confirm the process: you should obtain $\dfrac{2}{13} + \dfrac{29}{13} i.$