Which Expression Is Equivalent To -x² - 36? Options: A. (x+6i)(x-6i), B. (-x-6i)(x+6i), C. (-x+6i)(x-6i), D. (-x-6i)(-x+6i)

In the realm of algebra, expressions often hold hidden depths, waiting to be explored and understood. This article delves into a fascinating problem: identifying the expression that is equivalent to -x² - 36. We will dissect the given options, employing algebraic principles and complex number properties to arrive at the correct solution. This exploration will not only enhance your understanding of algebraic manipulations but also shed light on the intriguing interplay between real and imaginary numbers.

Decoding the Expression -x² - 36

To begin our quest, let's first dissect the expression -x² - 36. At first glance, it might appear to be a simple quadratic expression. However, the negative sign preceding the x² term and the constant term of -36 hint at a deeper structure. Our goal is to find an equivalent expression among the given options, which means we need to manipulate these options and see if they simplify to our target expression.

The expression -x² - 36 can be viewed as the negative of a sum of squares. Specifically, it is the negative of (x² + 36). The presence of the sum of squares is a crucial clue, as it suggests the involvement of complex numbers. Remember that complex numbers, which have both real and imaginary parts, often arise when dealing with square roots of negative numbers. This realization sets the stage for our investigation of the provided options.

Option A: (x + 6i)(x - 6i)

Let's examine the first contender, (x + 6i)(x - 6i). This expression is a product of two binomials, and it has a familiar form: the product of a sum and a difference. This pattern is known as the difference of squares, and it has a special property. When we multiply a sum and a difference of the same two terms, the result is the square of the first term minus the square of the second term. In mathematical notation, this is expressed as (a + b)(a - b) = a² - b².

Applying this principle to our expression, where a = x and b = 6i, we have:

(x + 6i)(x - 6i) = x² - (6i)²

Now, let's simplify further. We know that i is the imaginary unit, defined as the square root of -1. Therefore, i² = -1. Substituting this into our expression, we get:

x² - (6i)² = x² - (36 * i²) = x² - (36 * -1) = x² + 36

Notice that this result, x² + 36, is the opposite of our target expression, -x² - 36. Therefore, Option A is not the correct answer. However, this exercise highlights the importance of careful algebraic manipulation and the properties of complex numbers.

Option B: (-x - 6i)(x + 6i)

Moving on to the second option, we encounter (-x - 6i)(x + 6i). This expression might appear similar to the previous one, but the presence of the negative sign in front of the 'x' in the first binomial introduces a subtle yet significant difference. To unravel this expression, we'll employ the distributive property, carefully multiplying each term in the first binomial by each term in the second binomial.

Expanding the expression, we have:

(-x - 6i)(x + 6i) = -x * x + (-x) * 6i + (-6i) * x + (-6i) * 6i

Simplifying each term, we get:

-x² - 6xi - 6xi - 36i²

Combining like terms, we have:

-x² - 12xi - 36i²

Remember that i² = -1. Substituting this into our expression, we get:

-x² - 12xi - 36(-1) = -x² - 12xi + 36

This result, -x² - 12xi + 36, is clearly not equal to our target expression, -x² - 36, due to the presence of the imaginary term -12xi. Therefore, Option B is also incorrect. This reinforces the importance of meticulous expansion and simplification when dealing with complex expressions.

Option C: (-x + 6i)(x - 6i)

Let's consider the third option, (-x + 6i)(x - 6i). This expression, like Option B, involves complex numbers and requires careful expansion. We'll again use the distributive property to multiply the two binomials.

Expanding the expression, we get:

(-x + 6i)(x - 6i) = -x * x + (-x) * (-6i) + 6i * x + 6i * (-6i)

Simplifying each term, we have:

-x² + 6xi + 6xi - 36i²

Combining like terms, we get:

-x² + 12xi - 36i²

Substituting i² = -1, we have:

-x² + 12xi - 36(-1) = -x² + 12xi + 36

As with Option B, the presence of the imaginary term +12xi means that this expression, -x² + 12xi + 36, is not equivalent to our target expression, -x² - 36. Therefore, Option C is also incorrect.

Option D: (-x - 6i)(-x + 6i)

Finally, let's turn our attention to the fourth option, (-x - 6i)(-x + 6i). This expression, once again, presents a product of two binomials involving complex numbers. Notice that this expression also fits the pattern of the difference of squares, albeit with a slight twist. Here, we can consider a = -x and b = 6i. Applying the difference of squares formula, (a + b)(a - b) = a² - b², we have:

(-x - 6i)(-x + 6i) = (-x)² - (6i)²

Simplifying, we get:

x² - 36i²

Now, substituting i² = -1, we have:

x² - 36(-1) = x² + 36

Wait a minute! This result, x² + 36, is the same as what we obtained for Option A, and it's still not our target expression, -x² - 36. However, let's take a closer look. We made a mistake in the original response. We need the negative of this expression. Let's try factoring out a -1 from the original expression:

-x² - 36 = -(x² + 36)

Now, we need to find an option that equals -(x² + 36). Option D, as we calculated, results in x² + 36. To get the negative of this, we need to multiply the entire expression by -1. However, none of the options directly give us -(x² + 36). Let's re-examine our steps and the options carefully.

Reassessing the Options and the Target Expression

We've meticulously expanded and simplified each of the given options, and none of them directly match our target expression, -x² - 36. This suggests that we might need to re-evaluate our approach or consider alternative ways to manipulate the expressions. Let's revisit the options and our target expression with fresh eyes.

Our target expression, -x² - 36, can be rewritten as -(x² + 36). The term inside the parentheses, x² + 36, is a sum of squares, which, as we've seen, can be factored using complex numbers. However, the negative sign outside the parentheses adds a layer of complexity. We need to find an option that, when expanded and simplified, results in the negative of the sum of squares.

Let's revisit Option B: (-x - 6i)(x + 6i)

Expanding this again:

(-x - 6i)(x + 6i) = -x(x + 6i) - 6i(x + 6i) = -x² - 6xi - 6xi - 36i² = -x² - 12xi + 36

This still doesn't match. It seems there was an error in the provided options or the original question itself, as none of the options directly simplify to -x² - 36. However, if we are forced to choose the closest answer, we would reconsider option B.

Let's try factoring a -1 out of the first term in Option B:

(-x - 6i)(x + 6i) = -(x + 6i)(x + 6i) = -(x² + 12ix - 36)

This doesn't work either. Let's check Option D again.

Final Analysis and the Closest Possible Answer

After a thorough re-evaluation, it becomes evident that none of the provided options perfectly match the target expression, -x² - 36. This discrepancy suggests a potential error in the options themselves or the original question. However, in a situation where we must choose the closest possible answer, we need to analyze the expressions and identify the one that, with a minor modification, could potentially lead to the desired result.

We previously examined Option D, (-x - 6i)(-x + 6i), and found that it simplifies to x² + 36. This expression is the sum of squares, but it lacks the crucial negative sign that would make it equivalent to our target expression. However, if we were to multiply the entire expression by -1, we would obtain -(x² + 36), which is indeed equal to -x² - 36.

While none of the options directly equal -x² - 36, Option D, when multiplied by -1, gives us the correct expression. This indicates a possible missing negative sign in one of the factors. Given the constraints, we can conclude that Option D is the closest possible answer, even though it requires an additional step to perfectly match the target expression. This highlights the importance of meticulous algebraic manipulation and the ability to identify subtle differences in expressions.

Conclusion

In this exploration, we embarked on a journey to identify the expression equivalent to -x² - 36. We meticulously analyzed each of the given options, employing algebraic principles and the properties of complex numbers. While we discovered that none of the options perfectly matched our target expression, we were able to identify Option D, (-x - 6i)(-x + 6i), as the closest possible answer. This problem serves as a valuable reminder of the importance of careful algebraic manipulation, attention to detail, and the ability to recognize patterns and subtle differences in mathematical expressions.

Keywords: algebraic expressions, complex numbers, difference of squares, imaginary unit, quadratic expressions, equivalent expressions, -x² - 36