Question

select the correct answer. which expression is equivalent to the complex number 10 + 3i? a. 3i(4 + 7i) + (11 + 2i) b. 2i(4 - 5i) + (1 - 7i) c. (4 + 7i) - 2i(2 + 3i) d. (-3 + 5i) - 3i(4 + 5i)

Explanation:

Step1: Recall \(i^2 = -1\)

We will expand each option using the distributive property (FOIL for complex numbers) and simplify, then compare with \(10 + 3i\).

Step2: Simplify Option A

\[

$$\begin{align*} 3i(4 + 7i)+(11 + 2i)&=3i\times4+3i\times7i + 11 + 2i\\ &=12i + 21i^2+11 + 2i\\ &=12i+21(-1)+11 + 2i\\ &=12i - 21 + 11 + 2i\\ &=( - 21 + 11)+(12i + 2i)\\ &=-10 + 14i \end{align*}$$

\]
Not equal to \(10 + 3i\).

Step3: Simplify Option B

\[

$$\begin{align*} 2i(4 - 5i)+(1 - 7i)&=2i\times4-2i\times5i + 1 - 7i\\ &=8i-10i^2 + 1 - 7i\\ &=8i-10(-1)+1 - 7i\\ &=8i + 10 + 1 - 7i\\ &=(10 + 1)+(8i - 7i)\\ &=11 + i \end{align*}$$

\]
Not equal to \(10 + 3i\). Wait, maybe I made a mistake. Wait, let's re - calculate Option B:

Wait, \(2i(4 - 5i)=8i-10i^{2}\), since \(i^{2}=-1\), then \(8i - 10\times(-1)=8i + 10\). Then add \((1 - 7i)\): \(8i + 10+1 - 7i=(10 + 1)+(8i-7i)=11 + i\). Hmm, still not. Let's check Option C.

Step4: Simplify Option C

\[

$$\begin{align*} (4 + 7i)-2i(2 + 3i)&=4 + 7i-(4i+6i^{2})\\ &=4 + 7i - 4i-6i^{2}\\ &=4 + 7i - 4i-6\times(-1)\\ &=4 + 7i - 4i + 6\\ &=(4 + 6)+(7i - 4i)\\ &=10 + 3i \end{align*}$$

\]
This matches the given complex number \(10 + 3i\). Let's just check Option D for completeness.

Step5: Simplify Option D

\[

$$\begin{align*} (-3 + 5i)-3i(4 + 5i)&=-3 + 5i-(12i + 15i^{2})\\ &=-3 + 5i - 12i-15i^{2}\\ &=-3 + 5i - 12i-15\times(-1)\\ &=-3 + 5i - 12i + 15\\ &=(-3 + 15)+(5i - 12i)\\ &=12-7i \end{align*}$$

\]
Not equal to \(10 + 3i\).

Answer:

C. \((4 + 7i)-2i(2 + 3i)\)