Finding The Value Of 'c' For A Perfect Square Trinomial
In the realm of algebra, perfect square trinomials hold a special significance due to their predictable factorization. These trinomials, expressions with three terms, can be factored into the square of a binomial, making them invaluable in simplifying equations and solving problems. In this comprehensive exploration, we'll delve into the intricacies of perfect square trinomials, focusing on how to determine the value of a constant term that transforms a given expression into this specific form. We'll use a concrete example, the expression x² - 7x + c, and walk through the process of finding the value of c that makes it a perfect square trinomial. Understanding the principles behind this process will equip you with a powerful tool for algebraic manipulation and problem-solving.
Understanding Perfect Square Trinomials
To effectively tackle the problem at hand, it's crucial to first grasp the concept of perfect square trinomials. A perfect square trinomial is a trinomial that results from squaring a binomial. In simpler terms, it's an expression that can be factored into the form (ax + b)² or (ax - b)², where a and b are constants. Expanding these binomial squares reveals the characteristic structure of perfect square trinomials.
Consider the binomial (x + b)². Expanding this, we get:
(x + b)² = (x + b)(x + b) = x² + 2bx + b²
Similarly, for (x - b)², the expansion is:
(x - b)² = (x - b)(x - b) = x² - 2bx + b²
Notice the pattern that emerges in both expansions. A perfect square trinomial always consists of three terms: the square of the first term of the binomial (x² in this case), twice the product of the two terms of the binomial (2bx or -2bx), and the square of the second term of the binomial (b²). This pattern is the key to identifying and creating perfect square trinomials.
Recognizing this pattern allows us to work backward. If we're given a trinomial in the form x² + kx + c, we can determine whether it's a perfect square trinomial by checking if the constant term c is equal to the square of half the coefficient of the x term (which is k in this case). This relationship forms the foundation for solving our problem.
Identifying the Pattern
Now, let's focus on the expression x² - 7x + c. Our goal is to find the value of c that transforms this expression into a perfect square trinomial. To do this, we need to relate this expression to the general form of a perfect square trinomial, which we discussed earlier. Specifically, we'll use the form (x - b)² = x² - 2bx + b² since our expression has a negative sign in front of the x term.
Comparing x² - 7x + c with x² - 2bx + b², we can see a direct correspondence between the terms. The x² terms match, and the x terms give us a crucial piece of information: -7x = -2bx. This equation allows us to solve for b, which is essential for finding c.
Dividing both sides of the equation -7x = -2bx by -2x, we get:
b = 7/2
This tells us that the second term in our binomial is 7/2. Now, we need to find the constant term c that completes the perfect square trinomial. Recall that in the expanded form, the constant term is the square of b. Therefore, to find c, we simply need to square 7/2.
Calculating the Value of 'c'
To calculate the value of c, we square the value of b we just found, which is 7/2. This step directly applies the relationship we identified in the pattern of perfect square trinomials: the constant term is the square of half the coefficient of the x term.
So, we have:
c = (7/2)²
Squaring a fraction involves squaring both the numerator and the denominator. Therefore:
c = (7²)/(2²) = 49/4
This calculation reveals that the value of c that makes the expression x² - 7x + c a perfect square trinomial is 49/4. This result aligns with our understanding of the pattern and confirms our approach to solving the problem.
Therefore, the expression becomes x² - 7x + 49/4, which can be factored into (x - 7/2)². This confirms that we have indeed found the correct value of c that creates a perfect square trinomial.
Factoring the Perfect Square Trinomial
Having determined that c = 49/4 makes the expression a perfect square trinomial, let's explicitly factor the resulting trinomial to solidify our understanding. The expression we now have is x² - 7x + 49/4. As we discussed earlier, this should factor into the form (x - b)², where b is 7/2.
So, we can write:
x² - 7x + 49/4 = (x - 7/2)²
To verify this factorization, we can expand the right side of the equation:
(x - 7/2)² = (x - 7/2)(x - 7/2)
Using the distributive property (or the FOIL method), we get:
x² - (7/2)x - (7/2)x + (7/2)² = x² - 7x + 49/4
This confirms that our factorization is correct. The expression x² - 7x + 49/4 is indeed the square of the binomial (x - 7/2). This process of factoring reinforces the connection between perfect square trinomials and their binomial square roots.
Conclusion
In this exploration, we've successfully determined the value of c that transforms the expression x² - 7x + c into a perfect square trinomial. By understanding the pattern inherent in perfect square trinomials and applying algebraic principles, we found that c = 49/4 satisfies the condition. We also verified our result by factoring the resulting trinomial, solidifying our understanding of the relationship between perfect square trinomials and binomial squares.
The ability to identify and create perfect square trinomials is a valuable skill in algebra. It simplifies factoring, equation solving, and various other algebraic manipulations. By mastering this concept, you'll be well-equipped to tackle a wider range of mathematical problems with confidence. Remember the key pattern: in a perfect square trinomial of the form x² + kx + c, the constant term c is always the square of half the coefficient of the x term (k/2). This principle will guide you in identifying and manipulating these special trinomials effectively.
Choosing the Correct Answer
Based on our calculations and analysis, the value of c that makes the expression x² - 7x + c a perfect square trinomial is 49/4. Therefore, the correct answer from the given options is:
D. 49/4
This concludes our exploration of finding the value of c for a perfect square trinomial. We've not only arrived at the correct answer but also gained a deeper understanding of the underlying principles and their applications.
