Completing The Square: What Number To Add To Both Sides?

Hey guys! Let's dive into a common algebra problem: completing the square. This is a super useful technique for solving quadratic equations and understanding the standard form of a parabola. Today, we’re tackling the question: What number must be added to both sides of the equation $x^2 - 10x = 7$ to complete the square? Don't worry, we'll break it down step by step so it's crystal clear.

Understanding Completing the Square

Before we jump into the specific equation, let’s quickly recap what completing the square actually means. In essence, we're trying to transform a quadratic expression (like $x^2 - 10x$) into a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form $(x + a)^2$ or $(x - a)^2$. This makes it much easier to solve the equation or analyze the quadratic function. The main goal here is to manipulate the given equation into a form that reveals the vertex of the parabola represented by the quadratic equation. Knowing the vertex is crucial for graphing and understanding the behavior of the function. So, why is completing the square important? Well, it gives us a straightforward way to rewrite any quadratic equation in vertex form, which immediately tells us the vertex coordinates. This is super helpful in various applications, such as optimization problems where we need to find maximum or minimum values. Plus, it provides a reliable method for solving quadratic equations, especially when factoring isn't straightforward. Completing the square involves adding a specific value to both sides of the equation to maintain balance while creating the perfect square trinomial. This value is derived from the coefficient of the x term, and we'll see exactly how that works in our example. It's a fundamental algebraic technique with far-reaching applications. Understanding this method gives you a solid foundation for more advanced math concepts and problem-solving strategies. So, let's get into it and make sure we nail this technique down!

Step-by-Step Solution

Okay, let’s get our hands dirty with the equation $x^2 - 10x = 7$. Our mission is to figure out what magical number we need to add to both sides to complete the square. Here’s the breakdown:

1. Identify the Coefficient of the x Term

First things first, we need to pinpoint the coefficient of our $x$ term. In this equation, the $x$ term is $-10x$, so the coefficient is $-10$. This number is our key to unlocking the secret of completing the square. It’s like the main ingredient in our recipe for a perfect square trinomial. The coefficient of the x term tells us how much the binomial (x + a) or (x - a) will be adjusted to form the perfect square. Think of it as the linear adjustment needed to make the quadratic expression a perfect square.

2. Divide the Coefficient by 2

Next up, we take that coefficient (which is $-10$) and divide it by 2. So, $-10 / 2 = -5$. This step is crucial because it gives us the value that will be inside the parentheses when we write our perfect square trinomial in factored form. Dividing by 2 essentially finds the 'a' value in the (x - a)² or (x + a)² format. It's like finding the root of the perfect square, if you will. This intermediate value is pivotal in determining the final constant term that completes the square. You're essentially finding half of the x term's coefficient because the expansion of (x + a)² involves 2ax, so you need to reverse that process.

3. Square the Result

Now, we take the result from the previous step (which is $-5$) and square it. So, $(-5)^2 = 25$. This is the magic number we’ve been searching for! This value is what we need to add to both sides of the original equation to create a perfect square trinomial. Squaring the result ensures that we get a positive value, which is essential for completing the square. Think of it this way: squaring the result makes the constant term consistent with the binomial form we're aiming for. It’s the final touch that makes everything fit together perfectly. So, this 25 is crucial because it allows us to rewrite the left side of the equation as a squared binomial.

4. Add to Both Sides

Finally, we add 25 to both sides of the original equation to maintain the balance:

$x^2 - 10x + 25 = 7 + 25$

This is where the magic happens. Adding the same number to both sides keeps the equation balanced while transforming the left side into a perfect square. It's like adding the same weight to both sides of a scale to keep it even. This step is pivotal because it sets the stage for factoring the left side and solving for x. Now we have:

$x^2 - 10x + 25 = 32$

5. Factor and Simplify

The left side of the equation, $x^2 - 10x + 25$, is now a perfect square trinomial. We can factor it as $(x - 5)^2$. And on the right side, we have $7 + 25 = 32$. So, our equation becomes:

$(x - 5)^2 = 32$

The Answer

So, the number that must be added to both sides of the equation $x^2 - 10x = 7$ to complete the square is 25. See? Not too scary when we break it down step by step! This transformation is key to rewriting the quadratic equation in a form that makes it much easier to solve. The factored form highlights the value of x that makes the binomial equal to zero, which is a crucial step in finding the roots of the equation. Completing the square allows us to convert the quadratic into vertex form, which immediately reveals the vertex of the parabola represented by the equation. This is invaluable for graphing and understanding the behavior of the quadratic function.

Why Does This Work?

You might be wondering, why does this method of dividing by 2 and squaring actually work? Let’s break down the logic behind it. When we complete the square, we’re essentially trying to reverse the process of expanding a squared binomial. Consider the binomial $(x + a)^2$. When we expand this, we get:

$(x + a)^2 = x^2 + 2ax + a^2$

Notice that the coefficient of the $x$ term is $2a$, and the constant term is $a^2$. In our original equation, $x^2 - 10x = 7$, we had the $x^2$ and $x$ terms, but we were missing the constant term that would make it a perfect square. To find this constant term, we took half of the coefficient of the $x$ term (which gave us $a$) and then squared it (which gave us $a^2$). This is exactly the constant term we need to complete the square! So, the process we followed is a direct application of the expansion of a squared binomial, but in reverse. We’re using the coefficient of the x term to find the missing constant term that creates the perfect square trinomial. Understanding the underlying algebraic principle makes the technique much more intuitive and memorable. It’s not just a trick; it’s a carefully designed method that leverages the properties of squared binomials. This is why dividing by 2 and squaring the result works so effectively to complete the square.

Let's Practice!

Now that we’ve tackled this problem, let’s reinforce our understanding with a quick practice question. How would you complete the square for the equation $x^2 + 6x = 16$? Take a moment to work through the steps we discussed earlier. Identify the coefficient of the x term, divide it by 2, square the result, and add it to both sides. See if you can transform the equation into a perfect square trinomial. This practice is key to mastering the technique and making it second nature. Don't just read through the solution; actively engage with the problem and work it out for yourself. This will solidify your understanding and build your confidence. Think of each practice problem as a mini-challenge that helps you sharpen your skills. And remember, the more you practice, the easier it will become. So, give it a try, and let's make sure you've truly grasped the concept of completing the square. Once you've worked through this practice question, you'll be well on your way to becoming a pro at this essential algebraic technique.

Completing the square might seem tricky at first, but with a little practice, you'll get the hang of it. Remember the key steps: identify the coefficient of the $x$ term, divide by 2, square the result, and add to both sides. Keep practicing, and you’ll be a completing-the-square pro in no time!