Solving $x^2 + 8x + 7 = 0$ By Completing The Square A Step-by-Step Guide

Hey guys! Today, we're diving into a classic algebra problem: finding the $x$-intercepts of a quadratic equation. Specifically, we’ll be tackling the equation $x^2 + 8x + 7 = 0$ using the method of completing the square. This is a super useful technique, not just for solving quadratics, but also for understanding the structure and behavior of quadratic functions. So, let’s jump right in and make sure we nail this concept!

Understanding $x$-intercepts and Quadratic Equations

First off, let's make sure we're all on the same page about what $x$-intercepts actually are. In simple terms, the $x$-intercepts are the points where the graph of our quadratic equation crosses the $x$-axis. At these points, the value of $y$ (or the function value, which we often write as $f(x)$) is zero. That's why we set our quadratic equation equal to zero: to find the $x$ values that make the equation true, which are the $x$-intercepts.

Now, a quadratic equation is basically any equation that can be written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a$ isn't zero (otherwise, it wouldn't be quadratic!). Our equation, $x^2 + 8x + 7 = 0$, fits this form perfectly, with $a = 1$, $b = 8$, and $c = 7$.

There are several ways to solve quadratic equations – factoring, using the quadratic formula, and, the star of our show today, completing the square. Each method has its strengths, but completing the square is particularly insightful because it reveals the vertex form of the quadratic, which gives us a ton of information about the parabola (the U-shaped graph of a quadratic function). Plus, it's a foundational technique that pops up in all sorts of advanced math problems, so mastering it is a big win.

The Method: Completing the Square

Okay, let's break down the process of completing the square. It might seem a little tricky at first, but once you get the hang of it, it's like riding a bike. The main idea is to manipulate our quadratic equation into a form that includes a perfect square trinomial. A perfect square trinomial is a trinomial (that's a polynomial with three terms) that can be factored into the square of a binomial. For example, $x^2 + 6x + 9$ is a perfect square trinomial because it factors into $(x + 3)^2$.

Here’s the step-by-step guide to completing the square for our equation, $x^2 + 8x + 7 = 0$:

1. Isolate the $x^2$ and $x$ terms: We want to get the terms with $x$ on one side of the equation and the constant term on the other. So, we subtract 7 from both sides:

   $x^2 + 8x = -7$

2. Complete the square: This is the heart of the method. We need to add a number to both sides of the equation that will make the left side a perfect square trinomial. To find this number, we take half of the coefficient of our $x$ term (which is 8), square it, and add the result to both sides. Half of 8 is 4, and $4^2$ is 16. So, we add 16 to both sides:

   $x^2 + 8x + 16 = -7 + 16$

3. Factor the perfect square trinomial: The left side is now a perfect square trinomial! It factors beautifully into the square of a binomial. In our case, $x^2 + 8x + 16$ factors into $(x + 4)^2$. The right side simplifies to 9, so we have:

   $(x + 4)^2 = 9$

4. Take the square root of both sides: To get rid of the square on the left side, we take the square root of both sides of the equation. Remember, when we take the square root, we need to consider both the positive and negative roots:

   $x + 4 = \pm \sqrt{9}$

   $x + 4 = \pm 3$

5. Solve for $x$: We now have two simple linear equations to solve for $x$:

   • $x + 4 = 3$, which gives us $x = 3 - 4 = -1$
   • $x + 4 = -3$, which gives us $x = -3 - 4 = -7$

The Solution: $x$-intercepts Found!

And there we have it! We’ve found the $x$-intercepts of the quadratic equation $x^2 + 8x + 7 = 0$ by completing the square. Our solutions are $x = -1$ and $x = -7$. This corresponds to option (c) in the original question.

So, the correct answer is:

c. $x = -7, x = -1$

Visualizing the Solution

It’s always a good idea to connect our algebraic solution with a visual representation. If we were to graph the quadratic function $y = x^2 + 8x + 7$, we would see a parabola that opens upwards (since the coefficient of $x^2$ is positive). The parabola would cross the $x$-axis at the points $(-7, 0)$ and $(-1, 0)$, which are our $x$-intercepts. Knowing this helps us confirm that our solution makes sense in the context of the graph.

Why Completing the Square Matters

Completing the square isn't just a nifty trick for solving quadratic equations; it's a foundational technique that unlocks deeper understanding. Here’s why it’s so important:

• Vertex Form: Completing the square allows us to rewrite a quadratic equation in vertex form, which is $y = a(x - h)^2 + k$. The vertex form immediately tells us the vertex of the parabola, which is the point $(h, k)$. This is super useful for graphing and understanding the behavior of the quadratic function.

  In our example, after completing the square, we had $(x + 4)^2 = 9$. We can rewrite this as $y = (x + 4)^2 - 9$. So, the vertex of the parabola is $(-4, -9)$.

• Quadratic Formula Connection: The quadratic formula, which is a general solution for any quadratic equation, is actually derived by completing the square on the general quadratic equation $ax^2 + bx + c = 0$. So, mastering completing the square gives you a deeper appreciation for where the quadratic formula comes from.

• Calculus Applications: Completing the square is a technique that shows up in calculus, particularly when dealing with integrals involving quadratic expressions. Knowing how to manipulate quadratics in this way can simplify complex calculus problems.

Tips and Tricks for Mastering Completing the Square

Alright, guys, let's wrap up with some tips to help you become a completing-the-square pro:

• Practice, Practice, Practice: Like any math skill, the more you practice, the better you’ll get. Work through a variety of quadratic equations, and you’ll start to recognize patterns and shortcuts.

• Pay Attention to Signs: The signs are crucial when completing the square. Make sure you’re adding and subtracting the correct values, and be extra careful when taking the square root (remembering both positive and negative roots).

• Check Your Work: After you’ve found your solutions, plug them back into the original equation to make sure they work. This is a great way to catch any mistakes.

• Visualize the Process: Try graphing the quadratic function and identifying the $x$-intercepts visually. This will help you connect the algebraic solution with the graphical representation.

• Understand the “Why”: Don’t just memorize the steps. Understand why each step works and how it contributes to the overall goal of finding the $x$-intercepts. This deeper understanding will make you a more confident and capable problem-solver.

So, there you have it! We’ve successfully navigated the world of completing the square and found the $x$-intercepts of our quadratic equation. Keep practicing, keep exploring, and you’ll be a quadratic equation master in no time! You got this!