Completing The Square What Number To Add To Solve $x^2 + 8x = 4$

Hey guys! Ever stumbled upon a quadratic equation that looks like a jumbled mess? Don't sweat it! There's a nifty technique called "completing the square" that can help you solve these equations with ease. In this article, we'll break down this method, show you how to apply it, and answer that burning question: "What number should be added to both sides of the equation to complete the square?"

Understanding Quadratic Equations

Before we dive into the nitty-gritty, let's make sure we're all on the same page about quadratic equations. A quadratic equation is basically a polynomial equation of the second degree. That might sound like a mouthful, but it just means it has a term with $x^2$ in it. The standard form of a quadratic equation is:

$ax^2 + bx + c = 0$

Where 'a', 'b', and 'c' are constants, and 'a' can't be zero (otherwise, it wouldn't be a quadratic equation anymore!).

Now, solving quadratic equations can sometimes be tricky. You might be familiar with factoring or using the quadratic formula, which are great tools. But completing the square provides another powerful method, especially when the equation doesn't factor easily. It's also a fundamental technique used in deriving the quadratic formula itself, so understanding it gives you a deeper grasp of quadratic equations in general.

What is Completing the Square?

The core idea behind completing the square is to manipulate the quadratic equation into a perfect square trinomial on one side. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. Think of it like this:

$(x + k)^2 = x^2 + 2kx + k^2$

Notice how the constant term ($k^2$) is the square of half the coefficient of the x term (which is 2k). This relationship is the key to completing the square.

So, how do we use this to solve an equation? Let's say we have an equation that looks something like this:

$x^2 + bx = d$

Our goal is to add a constant to both sides of the equation so that the left side becomes a perfect square trinomial. To find that constant, we take half of the coefficient of the x term (which is b), square it, and add it to both sides. That is, we add $(b/2)^2$ to both sides of the equation.

Why Does This Work?

The magic happens because adding $(b/2)^2$ creates the perfect square trinomial we're after. The left side can then be factored into $(x + b/2)^2$, which simplifies the equation and allows us to solve for x.

Completing the square is a powerful technique because it allows us to rewrite a quadratic equation in a form where we can easily isolate x and find its value. It's like turning a complex puzzle into a simpler one, step by step.

Solving the Equation: ${ }^2 + 8x = 4$

Alright, let's tackle the specific equation you asked about: ${ }^2 + 8x = 4$. Notice that there seems to be a typo in the question. We'll assume it should be $x^2 + 8x = 4$. This is a classic example where completing the square shines.

Step 1: Identify the Coefficient of the x Term

In our equation, $x^2 + 8x = 4$, the coefficient of the x term is 8. This is the 'b' in our general form, which is key to finding the magic number we need to add.

Step 2: Calculate the Value to Add

Now, we need to figure out what number to add to both sides to make the left side a perfect square trinomial. Remember the rule? Take half of the coefficient of the x term and square it.

Half of 8 is 4, and 4 squared ($4^2$) is 16. So, the number we need to add is 16. This is the crucial step in completing the square.

Step 3: Add the Value to Both Sides

Now, add 16 to both sides of the equation:

$x^2 + 8x + 16 = 4 + 16$

This is where the magic really happens. By adding 16, we've transformed the left side into a perfect square trinomial.

Step 4: Factor the Perfect Square Trinomial

The left side, $x^2 + 8x + 16$, can be factored into $(x + 4)^2$. This is the whole point of completing the square: to rewrite the equation in this form.

Our equation now looks like this:

$(x + 4)^2 = 20$

See how much simpler it's become? We've turned a seemingly complex equation into something we can easily work with.

Step 5: Solve for x

To solve for x, we take the square root of both sides:

$x + 4 = \pm\sqrt{20}$

Remember, we need to consider both the positive and negative square roots.

Now, isolate x:

$x = -4 \pm \sqrt{20}$

We can simplify $\sqrt{20}$ as $\sqrt{4 * 5} = 2\sqrt{5}$, so our final solutions are:

$x = -4 + 2\sqrt{5}$ and $x = -4 - 2\sqrt{5}$

Answering the Question

So, to directly answer the question: What number should be added to both sides of the equation to complete the square? The answer is 16. You got it! This was the crucial step in transforming the equation into a solvable form.

Why 16 is the Key

The number 16 isn't just some random value we pulled out of thin air. It's precisely the number that completes the square because it makes the left side of the equation a perfect square trinomial. This allows us to factor it into the square of a binomial, which is the cornerstone of this method.

Adding 16 is the key to unlocking the solution because it allows us to rewrite the equation in a form where we can easily isolate x and find its value. It’s like adding the missing piece to a puzzle that makes everything fall into place.

The Answer Choices

Now, let's look at the answer choices you provided:

A. 4 B. 8 C. 16 D. 32

As we've demonstrated, the correct answer is C. 16. The other options might seem tempting, but they wouldn't result in a perfect square trinomial, and we wouldn't be able to solve the equation using this method.

Practice Makes Perfect

Completing the square might seem a bit daunting at first, but with practice, it becomes a powerful tool in your mathematical arsenal. The more you use it, the more comfortable you'll become with the process, and the quicker you'll be able to identify the number needed to complete the square.

Tips for Mastering Completing the Square

• Understand the Concept: Make sure you grasp the idea of perfect square trinomials and how they relate to binomial squares. This is the foundation of the method.
• Practice Regularly: The more you practice, the better you'll become at identifying the value to add and factoring the trinomial.
• Check Your Work: After completing the square, double-check that the trinomial you've created is indeed a perfect square. This will help you avoid errors.
• Use it in Different Contexts: Completing the square isn't just for solving equations. It's also used in other areas of math, like finding the vertex of a parabola or rewriting equations of circles. Explore these applications to deepen your understanding.

When to Use Completing the Square

Completing the square is a versatile technique, but it's particularly useful in certain situations:

• When the quadratic equation doesn't factor easily: If you've tried factoring and it's not working, completing the square is a great alternative.
• When you need to rewrite the equation in vertex form: The process of completing the square naturally leads to the vertex form of a quadratic equation, which is helpful for graphing and analyzing parabolas.
• When you want to derive the quadratic formula: As mentioned earlier, completing the square is the key to understanding where the quadratic formula comes from.

Completing the square offers a reliable method for finding solutions and gaining a deeper understanding of quadratic relationships.

Conclusion: Completing the Square, Conquered!

So, there you have it! We've walked through the process of completing the square, answered the question of what number to add to both sides (it's 16, by the way!), and explored why this method is so powerful. Remember, the key is to take half of the coefficient of the x term, square it, and add it to both sides. With a little practice, you'll be completing the square like a pro!

Don't be afraid to tackle those quadratic equations, guys. You've got this! Completing the square is a fantastic tool to have in your mathematical toolbox. Keep practicing, and you'll be amazed at what you can achieve. And remember, if you ever get stuck, just break it down step by step, and you'll get there. Happy solving!