Question

which of the following expressions is equal to $-x^2 - 36$?

a. $(-x + 6i)(x - 6i)$

b. $(-x - 6i)(x + 6i)$

c. $(-x - 6i)(x - 6i)$

d. $(x + 6i)(x - 6i)$

Explanation:

Step1: Recall the formula for \((a + b)(c + d)\)

The formula for multiplying two binomials \((a + b)(c + d)\) is \(ac+ad+bc+bd\). We will apply this formula to each option.

Step2: Analyze Option A \((-x + 6i)(x - 6i)\)

Using the distributive property (FOIL method):
\[

$$\begin{align*} (-x)(x)+(-x)(- 6i)+(6i)(x)+(6i)(-6i)&=-x^{2}+6ix + 6ix-36i^{2}\\ &=-x^{2}+12ix - 36i^{2} \end{align*}$$

\]
Since \(i^{2}=- 1\), we have \(-x^{2}+12ix+36\), which is not equal to \(-x^{2}-36\).

Step3: Analyze Option B \((-x - 6i)(x + 6i)\)

Using the distributive property:
\[

$$\begin{align*} (-x)(x)+(-x)(6i)+(-6i)(x)+(-6i)(6i)&=-x^{2}-6ix-6ix - 36i^{2}\\ &=-x^{2}-12ix-36i^{2} \end{align*}$$

\]
Substitute \(i^{2}=-1\): \(-x^{2}-12ix + 36\), which is not equal to \(-x^{2}-36\).

Step4: Analyze Option C \((-x - 6i)(x - 6i)\)

Using the distributive property:
\[

$$\begin{align*} (-x)(x)+(-x)(-6i)+(-6i)(x)+(-6i)(-6i)&=-x^{2}+6ix-6ix + 36i^{2}\\ &=-x^{2}+36i^{2} \end{align*}$$

\]
Substitute \(i^{2}=-1\): \(-x^{2}-36\), which is equal to the given expression.

Step5: Analyze Option D \((x + 6i)(x - 6i)\)

This is a difference of squares: \(x^{2}-(6i)^{2}=x^{2}-36i^{2}\)
Substitute \(i^{2}=-1\): \(x^{2}+36\), which is not equal to \(-x^{2}-36\).

Answer:

C. \((-x - 6i)(x - 6i)\)