Solve Quadratics By Completing The Square

Hey guys! Today, we're diving into a super useful technique in algebra: completing the square. This method is a fantastic way to solve quadratic equations, and it's especially handy when factoring isn't straightforward. We'll break down the process step-by-step, making sure you understand each part. Let's use the equation $x^2 + 12x = -20$ as our example. So, grab your pencils, and let's get started!

1. Understanding the Goal of Completing the Square

Before we jump into the nitty-gritty, let's quickly discuss the main goal of completing the square. The whole idea is to transform a quadratic equation in the form of $ax^2 + bx + c = 0$ into a perfect square trinomial. What's a perfect square trinomial, you ask? It's a trinomial that can be factored into the square of a binomial, like $(x + k)^2$ or $(x - k)^2$. When we have an equation in this form, solving for x becomes much easier because we can use the square root property. For our example equation, $x^2 + 12x = -20$, we want to manipulate the left side into a perfect square trinomial. This involves adding a specific value to both sides of the equation, which we'll figure out in the next step. So, remember, the ultimate aim is to rewrite the quadratic expression as a squared binomial, making it simpler to solve. This method not only helps in solving equations but also in understanding the structure and properties of quadratic functions. Keep this goal in mind as we move through the steps, and you'll find the process much more intuitive!

2. Finding the Value to Complete the Square

Okay, so how do we figure out what magical number to add? Here's the secret formula: $\left(\frac{b}{2}\right)^2$. This little expression is the key to completing the square. But what's b? Well, in our quadratic equation, which is in the form $ax^2 + bx + c = 0$, b is the coefficient of the x term. In our example, $x^2 + 12x = -20$, the b value is 12. Now, let's plug that into our formula: $\left(\frac{12}{2}\right)^2$. First, we divide 12 by 2, which gives us 6. Then, we square 6, which gives us 36. Ta-da! The value we need to add to both sides of the equation to complete the square is 36. Make sure you understand where this number comes from, guys. It's not just a random number; it's calculated specifically to make the left side of the equation a perfect square trinomial. Understanding this step is crucial, as it forms the foundation for the rest of the process. So, remember the formula $\left(\frac{b}{2}\right)^2$, identify b correctly, and you're golden!

3. Adding $(\frac{b}{2})^2$ to Both Sides

Alright, now that we've found our magic number, 36, it's time to put it to work. The next step is adding this value to both sides of our equation. Remember, whatever we do to one side of an equation, we must do to the other side to keep it balanced. So, let's take our equation, $x^2 + 12x = -20$, and add 36 to both sides. This gives us: $x^2 + 12x + 36 = -20 + 36$. Now, let's simplify the right side of the equation. -20 plus 36 equals 16. So, our equation now looks like this: $x^2 + 12x + 36 = 16$. See how the left side now has three terms? This is our perfect square trinomial in the making! The act of adding the same value to both sides is a fundamental principle in algebra, ensuring that the equation remains equivalent to its original form. This step is not just about adding a number; it's about strategically transforming the equation into a form that's easier to solve. By adding 36, we've set the stage for the next crucial step: writing the left side as a binomial squared.

4. Writing the Left Side as a Binomial Squared

This is where the magic truly happens! Now, we're going to rewrite the left side of our equation, $x^2 + 12x + 36$, as a binomial squared. Remember, a perfect square trinomial can always be factored into the form $(x + k)^2$ or $(x - k)^2$. So, how do we figure out what k is in our case? Well, k is simply half of the coefficient of our x term (the b value) before we squared it. Remember when we calculated $\frac{b}{2}$? That's our k! In our equation, $b$ is 12, so $\frac{12}{2}$ is 6. This means our binomial squared will be $(x + 6)^2$. Let's check if this is correct by expanding $(x + 6)^2$. Using the FOIL method (First, Outer, Inner, Last), we get: $(x + 6)(x + 6) = x^2 + 6x + 6x + 36 = x^2 + 12x + 36$. It matches the left side of our equation! Awesome! So, we can rewrite our equation $x^2 + 12x + 36 = 16$ as $(x + 6)^2 = 16$. This step is super important because it simplifies the equation into a form where we can easily isolate x. By recognizing the perfect square trinomial and factoring it into a binomial squared, we've made the equation much more manageable.

5. Solving for x Using the Square Root Property

We're in the home stretch now, guys! We've transformed our equation into $(x + 6)^2 = 16$. Now, we're going to use the square root property to solve for x. The square root property states that if $a^2 = b$, then $a = \pm\sqrt{b}$. This means we need to take the square root of both sides of our equation. When we take the square root of $(x + 6)^2$, we get $x + 6$. And when we take the square root of 16, we get both positive and negative 4 (because both $4^2$ and $(-4)^2$ equal 16). So, we have $x + 6 = \pm 4$. Now, we have two separate equations to solve: 1. $x + 6 = 4$ 2. $x + 6 = -4$ Let's solve the first equation: $x + 6 = 4$. To isolate x, we subtract 6 from both sides: $x = 4 - 6$, which gives us $x = -2$. Now, let's solve the second equation: $x + 6 = -4$. Again, we subtract 6 from both sides: $x = -4 - 6$, which gives us $x = -10$. So, our solutions are $x = -2$ and $x = -10$. Remember, quadratic equations often have two solutions, and we've found both of them by using the square root property. This step is crucial for finding the actual values of x that satisfy the original equation. By understanding and applying the square root property, we can effectively solve quadratic equations that are in the form of a binomial squared.

6. Checking Your Solutions

It's always a good idea to check your answers to make sure they're correct! To do this, we'll plug each solution back into our original equation, $x^2 + 12x = -20$, and see if it holds true. Let's start with $x = -2$: $(-2)^2 + 12(-2) = -20$ $4 - 24 = -20$ $-20 = -20$ It checks out! Now, let's check $x = -10$: $(-10)^2 + 12(-10) = -20$ $100 - 120 = -20$ $-20 = -20$ It checks out too! Both of our solutions are correct. Checking your solutions is a vital step in the problem-solving process. It helps you catch any potential errors and ensures that your answers are accurate. By plugging the solutions back into the original equation, we verify that they indeed satisfy the equation, giving us confidence in our results. So, always remember to check your solutions whenever you solve an equation!

Conclusion: Mastering Completing the Square

And there you have it! We've successfully solved the quadratic equation $x^2 + 12x = -20$ by completing the square. We found that the solutions are $x = -2$ and $x = -10$. Remember, the key steps are: 1. Finding the value to complete the square using the formula $\left(\frac{b}{2}\right)^2$. 2. Adding that value to both sides of the equation. 3. Writing the left side as a binomial squared. 4. Solving for x using the square root property. 5. Checking your solutions. Completing the square might seem tricky at first, but with practice, it becomes a powerful tool in your algebra arsenal. Not only does it help you solve quadratic equations, but it also provides a deeper understanding of their structure. So, keep practicing, and you'll master this technique in no time! If you have any questions, feel free to ask. Happy solving, guys!