Rewriting X^2-12x+49 In Vertex Form By Completing The Square

To rewrite the quadratic expression $x^2 - 12x + 49$ in vertex form by completing the square, we'll walk through the process step-by-step. Vertex form is a fantastic way to express quadratic equations because it immediately reveals the vertex (the minimum or maximum point) of the parabola. Guys, this method is super useful in algebra and beyond, so let’s dive in!

Understanding Vertex Form

Before we start, let's clarify what vertex form actually is. A quadratic equation in vertex form looks like this:

$f(x) = a(x - h)^2 + k$

Where:

• a determines the direction and stretch of the parabola.
• (h, k) represents the vertex of the parabola. The vertex is the point where the parabola changes direction (either the minimum or maximum point).

Our goal is to transform $x^2 - 12x + 49$ into this beautiful form. Completing the square is the technique we'll use to get there.

Step-by-Step: Completing the Square

Step 1: Focus on the Quadratic and Linear Terms

First, we isolate the terms with $x^2$ and $x$. In our expression, $x^2 - 12x + 49$, we focus on $x^2 - 12x$. The constant term, 49, will come into play later. Think of it like we're setting up a perfect square trinomial. This is crucial to understand the underlying principle of the method.

Step 2: Find the Value to Complete the Square

The core of completing the square lies in finding a constant that, when added to $x^2 - 12x$, creates a perfect square trinomial. A perfect square trinomial can be factored into the form $(x + b)^2$ or $(x - b)^2$. To find this constant, we take half of the coefficient of the $x$ term and square it. In our case, the coefficient of the $x$ term is -12.

• Half of -12 is -6.
• Squaring -6 gives us $(-6)^2 = 36$.

So, 36 is the magic number that will complete the square!

Step 3: Add and Subtract the Value

Now, we add and subtract 36 within the expression. Why add and subtract? Because adding a value changes the expression, but adding and subtracting the same value is equivalent to adding 0, which keeps the expression mathematically the same. This is a key trick in algebra. Our expression now looks like this:

$x^2 - 12x + 36 - 36 + 49$

Step 4: Factor the Perfect Square Trinomial

The first three terms, $x^2 - 12x + 36$, form a perfect square trinomial. We can factor this into $(x - 6)^2$. Remember, we found 36 by squaring half of -12, so this factorization should come naturally. Guys, this is where all the hard work pays off!

Our expression now looks like:

$(x - 6)^2 - 36 + 49$

Step 5: Simplify the Constant Terms

Finally, we combine the constant terms: -36 + 49 = 13. This gives us the vertex form:

$(x - 6)^2 + 13$

The Vertex Form Revealed

We've successfully rewritten $x^2 - 12x + 49$ in vertex form: $(x - 6)^2 + 13$. From this form, we can immediately identify the vertex of the parabola. Comparing our result to the general vertex form $a(x - h)^2 + k$, we see that:

• a = 1 (which means the parabola opens upwards)
• h = 6
• k = 13

Therefore, the vertex of the parabola is (6, 13). This means the parabola has a minimum point at (6, 13). Understanding this is super important for graphing quadratic functions and solving related problems.

Why is Vertex Form Useful?

Vertex form isn't just a mathematical curiosity; it's incredibly practical. Here’s why:

1. Identifying the Vertex: As we’ve seen, vertex form directly reveals the vertex of the parabola, which is crucial for understanding the function’s behavior.
2. Finding Maximum or Minimum Values: The vertex represents the maximum or minimum value of the quadratic function. If a is positive, the parabola opens upwards, and the vertex is the minimum point. If a is negative, the parabola opens downwards, and the vertex is the maximum point. This is super useful in optimization problems.
3. Graphing Quadratic Functions: Vertex form makes it easy to graph quadratic functions. You know the vertex, and the a value tells you whether the parabola opens up or down and how stretched or compressed it is. This makes graphing much simpler.

Common Mistakes to Avoid

Completing the square can be a bit tricky at first, so let's look at some common mistakes to avoid:

• Forgetting to Add and Subtract: Remember, you need to add and subtract the value you calculated to complete the square. Otherwise, you’re changing the equation.
• Incorrectly Factoring: Make sure you factor the perfect square trinomial correctly. Double-check your work!
• Misidentifying the Vertex: The vertex is (h, k), not (-h, k). Pay attention to the signs in the vertex form equation. This is a very common mistake, so be careful!

Practice Makes Perfect

The best way to master completing the square is through practice. Try rewriting different quadratic expressions in vertex form. Start with simpler examples and gradually move to more complex ones. Guys, the more you practice, the easier it will become!

Examples

Let's look at a couple more examples to solidify our understanding.

Example 1: Rewrite $x^2 + 8x + 10$ in vertex form

1. Focus: $x^2 + 8x$
2. Find the Value: Half of 8 is 4, and $4^2 = 16$
3. Add and Subtract: $x^2 + 8x + 16 - 16 + 10$
4. Factor: $(x + 4)^2 - 16 + 10$
5. Simplify: $(x + 4)^2 - 6$

The vertex form is $(x + 4)^2 - 6$, and the vertex is (-4, -6).

Example 2: Rewrite $2x^2 - 16x + 25$ in vertex form

This one is a bit trickier because we have a coefficient in front of the $x^2$ term. Here's how to handle it:

1. Factor out the coefficient: $2(x^2 - 8x) + 25$
2. Focus: $x^2 - 8x$
3. Find the Value: Half of -8 is -4, and $(-4)^2 = 16$
4. Add and Subtract (inside the parentheses): $2(x^2 - 8x + 16 - 16) + 25$
5. Factor: $2((x - 4)^2 - 16) + 25$
6. Distribute the 2: $2(x - 4)^2 - 32 + 25$
7. Simplify: $2(x - 4)^2 - 7$

The vertex form is $2(x - 4)^2 - 7$, and the vertex is (4, -7). This example highlights the importance of handling coefficients correctly. Failing to do so can lead to major errors.

Conclusion

Completing the square is a powerful technique for rewriting quadratic expressions in vertex form. It allows us to easily identify the vertex of the parabola and understand the function's behavior. While it might seem challenging at first, with practice, you'll become a pro at it. Remember to focus on the steps, avoid common mistakes, and most importantly, have fun with it! Understanding these concepts is essential for success in algebra and calculus. Keep practicing, guys, and you’ll master it in no time!