Question

what values of \\(c\\) and \\(d\\) would make the following expression represent a real number?
\\(i(2+3i)(c+di)\\)

• \\(c = 2, d = 3\\)
• \\(c = -2, d = 3\\)
• \\(c = 3, d = -2\\)
• \\(c = -3, d = -2\\)

Explanation:

🆕 New Concept Discovered: Operations with Complex Numbers
Working with the imaginary unit \( i \) where \( i^2 = -1 \).

Step 1: Correct the expression typo

Looking closely at the image, the expression is written as:

$$ i(2 + 3)(c + d) $$

However, in standard complex number problems of this type, the factor \( (2+3) \) is a typo for the complex number \( (2 + 3i) \), and the factor \( (c+d) \) represents \( (c + di) \). Let's write the intended complex expression:

$$ i(2 + 3i)(c + di) $$

Step 2: Expand the expression

First, distribute the outer \( i \) into the first binomial:

$$ i(2 + 3i) = 2i + 3i^2 $$

Since \( i^2 = -1 \):

$$ 2i + 3(-1) = -3 + 2i $$

Now, multiply this result by the second binomial \( (c + di) \):

$$ (-3 + 2i)(c + di) $$

Using the FOIL method to expand:

$$ (-3)(c) + (-3)(di) + (2i)(c) + (2i)(di) $$

$$ = -3c - 3di + 2ci + 2di^2 $$

Substitute \( i^2 = -1 \):

$$ = -3c - 3di + 2ci - 2d $$

Step 3: Group into real and imaginary parts

Group the terms without \( i \) (real part) and the terms with \( i \) (imaginary part):

$$ \text{Real part: } -3c - 2d $$

$$ \text{Imaginary part: } (2c - 3d)i $$

So, the fully expanded expression is:

$$ (-3c - 2d) + (2c - 3d)i $$

Step 4: Set the imaginary part to zero

For the expression to represent a pure real number, the imaginary part must equal zero:

$$ 2c - 3d = 0 $$

Now, we test the given multiple-choice options to see which pair of \( c \) and \( d \) satisfies this equation:

• Option 1: \( c = 2, d = 3 \)

$$ 2(2) - 3(3) = 4 - 9 = -5 eq 0 $$

• Option 2: \( c = -2, d = 3 \)

$$ 2(-2) - 3(3) = -4 - 9 = -13 eq 0 $$

• Option 3: \( c = 3, d = -2 \)

$$ 2(3) - 3(-2) = 6 + 6 = 12 eq 0 $$

• Option 4: \( c = -3, d = -2 \)

$$ 2(-3) - 3(-2) = -6 + 6 = 0 $$

The fourth option satisfies the condition perfectly.

Answer:

\( c = -3, d = -2 \)