Perfect Square Trinomial: Find The Value Of 'c'
Have you ever stumbled upon a quadratic expression and wondered what it takes to transform it into a perfect square trinomial? Well, you're in the right place! Let's dive deep into the world of quadratic expressions and explore the fascinating concept of perfect square trinomials. In this article, we'll tackle the question: What value of 'c' makes the expression x² - 24x + c a perfect square trinomial? We'll break down the process step by step, ensuring you grasp the underlying principles and can confidently solve similar problems.
What is a Perfect Square Trinomial?
Before we jump into solving for 'c', let's first define what a perfect square trinomial actually is. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. In simpler terms, it's a quadratic expression that results from squaring a binomial. Think of it like this:
(ax + b)² = (ax + b)(ax + b) = a²x² + 2abx + b²
Notice the pattern? A perfect square trinomial has a specific structure:
- The first term is a perfect square (a²x²).
- The last term is a perfect square (b²).
- The middle term is twice the product of the square roots of the first and last terms (2abx).
Understanding this pattern is key to identifying and creating perfect square trinomials. Now, let’s move on to the core of our problem.
The Quest for 'c': Completing the Square
Our mission is to find the value of 'c' that transforms x² - 24x + c into a perfect square trinomial. To achieve this, we'll employ a technique called completing the square. This method allows us to manipulate a quadratic expression into the perfect square form. Here's how it works:
- Focus on the coefficient of the 'x' term: In our expression, x² - 24x + c, the coefficient of the 'x' term is -24. This is the crucial number we'll be working with.
- Divide the coefficient by 2: Divide -24 by 2, which gives us -12.
- Square the result: Square -12, which yields (-12)² = 144.
The value we just calculated, 144, is the value of 'c' that will make our expression a perfect square trinomial! But why does this work? Let's break it down further.
Unveiling the Magic: Why Completing the Square Works
The process of completing the square is rooted in the algebraic identity of a squared binomial. Remember the pattern we discussed earlier:
(ax + b)² = a²x² + 2abx + b²
In our case, we have x² - 24x + c. We want to find a value for 'c' that allows us to rewrite this expression in the form (x + b)². Let's expand (x + b)²:
(x + b)² = x² + 2bx + b²
Now, let's compare this to our original expression, x² - 24x + c. We can see that:
- The coefficient of the 'x' term in (x + b)² is 2b, which corresponds to -24 in our expression.
- The constant term in (x + b)² is b², which corresponds to 'c' in our expression.
From the first point, we have 2b = -24. Dividing both sides by 2, we get b = -12. Now, let's substitute this value of 'b' into the second point:
c = b² = (-12)² = 144
This confirms our earlier calculation! When c = 144, our expression becomes a perfect square trinomial.
Putting it All Together: The Perfect Square Trinomial
Now that we've found the value of 'c', let's rewrite our expression:
x² - 24x + 144
Can we factor this into the square of a binomial? Absolutely! We know that b = -12, so we can write:
x² - 24x + 144 = (x - 12)²
This clearly demonstrates that x² - 24x + 144 is indeed a perfect square trinomial. We have successfully completed the square and found the value of 'c'.
The Answer and Its Significance
Therefore, the value of 'c' that makes x² - 24x + c a perfect square trinomial is 144. This corresponds to option D in the original question.
Understanding perfect square trinomials and the technique of completing the square is not just about solving this specific problem. It's a fundamental concept in algebra that has wide-ranging applications, including:
- Solving quadratic equations: Completing the square is a powerful method for finding the roots of quadratic equations.
- Graphing quadratic functions: The vertex form of a quadratic equation, which is derived from completing the square, makes it easy to identify the vertex and axis of symmetry of the parabola.
- Calculus: Completing the square is used in various calculus techniques, such as integration.
By mastering this concept, you're equipping yourself with a valuable tool for tackling a variety of mathematical challenges.
Practice Makes Perfect: Examples and Exercises
To solidify your understanding, let's explore a few more examples and exercises.
Example 1:
Find the value of 'k' that makes x² + 10x + k a perfect square trinomial.
- Coefficient of 'x' term: 10
- Divide by 2: 10 / 2 = 5
- Square the result: 5² = 25
Therefore, k = 25, and the perfect square trinomial is x² + 10x + 25 = (x + 5)²
Example 2:
Find the value of 'm' that makes x² - 6x + m a perfect square trinomial.
- Coefficient of 'x' term: -6
- Divide by 2: -6 / 2 = -3
- Square the result: (-3)² = 9
Therefore, m = 9, and the perfect square trinomial is x² - 6x + 9 = (x - 3)²
Exercises:
- Find the value of 'n' that makes x² + 16x + n a perfect square trinomial.
- Find the value of 'p' that makes x² - 20x + p a perfect square trinomial.
- Find the value of 'q' that makes x² + 7x + q a perfect square trinomial. (Hint: You may need to work with fractions here!)
Try solving these exercises to further reinforce your understanding of completing the square.
Conclusion: Mastering Perfect Square Trinomials
In this article, we've embarked on a journey to understand perfect square trinomials and the method of completing the square. We've discovered how to find the value of 'c' that transforms a quadratic expression into a perfect square trinomial. By understanding the underlying principles and practicing with examples, you can confidently tackle these types of problems and apply this knowledge to various mathematical contexts.
Remember, mathematics is a journey of exploration and discovery. Keep practicing, keep asking questions, and keep pushing your understanding further. The world of perfect square trinomials is now a little less mysterious, and you're well-equipped to navigate it!
For further exploration and a deeper dive into quadratic equations and related concepts, check out resources like Khan Academy's Algebra 1 section.
