Equivalent Expression For (3x - √7i)²

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Polynomial expressions form a cornerstone of algebraic manipulation, and mastering their transformations is crucial for success in mathematics. In this article, we will dive deep into the intricacies of one such transformation, focusing on the expression (3x - √7i)². Our primary goal is to identify the equivalent expression from a set of options, which requires a thorough understanding of complex numbers, binomial expansion, and algebraic manipulation. Let's embark on this journey, unraveling the complexities and arriving at the correct solution.

To effectively tackle this problem, we need to break it down into manageable steps. First, we need to recognize the structure of the given expression. It is a binomial, (3x - √7i), squared. This immediately suggests the application of the binomial expansion formula or, more specifically, the formula for the square of a difference. This formula states that (a - b)² = a² - 2ab + b². By applying this formula, we can expand the given expression and simplify it. This is where a solid grasp of algebraic identities becomes indispensable. These identities are the tools that allow us to navigate the algebraic terrain with confidence and precision.

Once we've expanded the expression, the next step involves dealing with the imaginary unit, 'i'. Remember that i is defined as the square root of -1, and therefore, i² = -1. This seemingly simple identity is the key to simplifying terms involving 'i' and is essential for reducing the expression to its simplest form. Ignoring this step can lead to incorrect results, highlighting the importance of a meticulous approach. This principle is crucial in complex number arithmetic and is applied extensively in various fields of mathematics and engineering.

Now, let's apply these principles to the given expression. The first step is to square the first term, (3x). Squaring 3x results in 9x². This is a straightforward application of the power rule. Next, we multiply the two terms, 3x and -√7i, and double the result. This gives us 2 * (3x) * (-√7i) = -6√7xi. Finally, we square the second term, (-√7i). This yields (-√7i)² = 7i². But remember, i² = -1, so this term becomes -7. Now we can assemble the expanded expression: 9x² - 6√7xi - 7. This expanded form is the result of applying the binomial expansion and simplifying using the properties of complex numbers.

The expanded expression, 9x² - 6√7xi - 7, doesn't directly match any of the provided options. This is where the critical step of recognizing complex conjugates comes into play. The original expression, (3x - √7i)², suggests a connection to expressions involving the conjugate of (3x - √7i), which is (3x + √7i). The options presented likely aim to test our understanding of how conjugates interact when multiplied or squared. Recognizing this pattern is crucial for navigating the problem efficiently and accurately.

To determine the equivalent expression, we should examine the options more closely. Let's start with the option (3x + √7i)(3x - √7i). This expression is the product of a complex number and its conjugate. When a complex number is multiplied by its conjugate, the result is always a real number. Specifically, (a + bi)(a - bi) = a² + b². Applying this to our expression, we get (3x)² + (√7)² = 9x² + 7. This result is not equivalent to our expanded expression, 9x² - 6√7xi - 7. However, it does provide valuable insight into the behavior of complex conjugates.

Next, let's consider the option (3x + √7i)(3x - √7i). As we've already discussed, this is the product of a complex number and its conjugate. Applying the difference of squares formula, (a + b)(a - b) = a² - b², we get (3x)² - (√7i)² = 9x² - 7i². Since i² = -1, this simplifies to 9x² + 7. Again, this result does not match our expanded expression, indicating that this option is incorrect.

Now, let's analyze the option (3x + √7i)(3x - √7i). This expression involves the product of two binomials. Multiplying these binomials, we get 9x² - 3x√7i + 3x√7i - 7i². The middle terms cancel out, and we are left with 9x² - 7i². Since i² = -1, this simplifies to 9x² + 7. This result is different from our expanded form, suggesting that this is not the equivalent expression we are looking for.

Finally, let's examine the option (3x + √7)(3x - √7). This is a difference of squares, where a = 3x and b = √7. Applying the difference of squares formula, (a + b)(a - b) = a² - b², we get (3x)² - (√7)² = 9x² - 7. This expression is a real number and does not contain any imaginary terms, making it distinct from our expanded expression which contains an imaginary term. Therefore, this option is also incorrect.

Therefore, after careful expansion, simplification, and comparison, the expression (3x - √7i)² is equivalent to 9x² - 6√7xi - 7. Recognizing the application of the binomial theorem and the properties of complex numbers were essential steps in solving this problem. By methodically working through the expansion and simplification process, we can arrive at the correct equivalent expression. This exercise underscores the significance of a strong foundation in algebraic manipulation and complex number theory for tackling polynomial problems successfully.

Now, let’s meticulously analyze each of the given options to pinpoint the correct equivalent expression for (3x - √7i)². This step-by-step approach will not only lead us to the solution but also enhance our understanding of complex number manipulation and polynomial expressions.

  1. (3x - √7i)²

    • This is the original expression that we need to expand and simplify. Applying the formula for the square of a binomial, (a - b)² = a² - 2ab + b², where a = 3x and b = √7i, we get:
    • (3x)² - 2(3x)(√7i) + (√7i)² = 9x² - 6√7xi + 7i²
    • Since i² = -1, we can substitute this into the expression: 9x² - 6√7xi - 7
    • This expanded form serves as our benchmark for comparison with the other options.

  2. (3x + √7)²

    • Expanding this expression using the formula for the square of a binomial, (a + b)² = a² + 2ab + b², where a = 3x and b = √7, we get:
    • (3x)² + 2(3x)(√7) + (√7)² = 9x² + 6√7x + 7
    • This expression is a real number polynomial, and it doesn't have any imaginary part. It is clearly different from the expanded form of the original expression (9x² - 6√7xi - 7).

  3. (3x + √7)(3x - √7)

    • This expression is in the form of the difference of squares, (a + b)(a - b) = a² - b², where a = 3x and b = √7.
    • Applying the formula, we get: (3x)² - (√7)² = 9x² - 7
    • This is also a real number polynomial, lacking the imaginary component present in the expanded original expression. Therefore, it is not equivalent.

  4. (3x + √7i)(3x - √7i)

    • This expression represents the product of a complex number and its conjugate. Using the difference of squares formula, (a + bi)(a - bi) = a² + b², where a = 3x and b = √7, we get:
    • (3x)² - (√7i)² = 9x² - 7i²
    • Since i² = -1, we substitute to get: 9x² - 7(-1) = 9x² + 7
    • Again, this is a real number polynomial, and it does not match the imaginary part of our original expanded form.

After thoroughly expanding and simplifying the original expression (3x - √7i)² and comparing it with each of the provided options, we can conclusively determine the equivalent expression. The original expression expands to 9x² - 6√7xi - 7, which explicitly includes a real part (9x² - 7) and an imaginary part (-6√7xi). This expanded form is the result of applying the binomial expansion formula and the property of the imaginary unit, i² = -1.

The systematic analysis of each option reveals that none of them directly match the expanded form of the original expression. The options (3x + √7)², (3x + √7)(3x - √7), and (3x + √7i)(3x - √7i) all result in real number polynomials, devoid of any imaginary components. This divergence underscores the importance of accurately handling complex numbers and their conjugates in algebraic manipulations.

Therefore, based on our detailed examination, we can confidently conclude that none of the provided options is precisely equivalent to the expanded form of the original expression, which is 9x² - 6√7xi - 7. This outcome highlights the significance of meticulous expansion and simplification when dealing with complex number expressions. The careful application of algebraic principles and a keen understanding of complex number properties are essential for solving such problems accurately.

Therefore, based on our detailed examination, we can confidently conclude that the correct answer is 9x² - 6√7xi - 7