It’s pretty much a guarantee that you’ll see quadratic equations on the SAT and ACT. But they can be tricky to tackle, especially since there are multiple methods you can use to solve them.
In this article, we’re going to walk through using one specific method—completing the square—to solve a quadratic equation. In fact, we’ll give you step-by-step instructions for how to complete the square using the completing the square formula. By the end, you should have a better understanding of how and when to use this mathematical strategy!
Ready to learn more? Then let’s jump in!
Engineers use quadratic equations to design roller coasters!
What Is a Quadratic Equation?
In order to understand how to complete the square, you first have to know how to identify a quadratic equation. That’s because completing the square only applies to quadratic equations!
In math, a quadratic equation is any equation that has the following formula:
$ax^2 + bx + c = 0$
In this equation, $x$ represents an unknown number and $a$ cannot be 0. (If $a$ is 0, then the equation is linear, not quadratic!)
Quadratic equations have all sorts of real-world applications because they're used to calculate parabolas, or arcs. Construction projects like bridges use the quadratic equation to calculate the arc of the structure, and even roller coasters use quadratics to design adrenaline-pumping tracks. Quadratics even fuel popular video games like Angry Birds, where the arc of each bird is calculated using the quadratic formula!
So now that you know why quadratic equations are important, let’s look at one of the most common methods of solving them: completing the square.
What Is Completing The Square and When Do You Use It?
There are actually four ways to solve a quadratic equation: taking the square root, factoring, completing the square, and the quadratic formula. Unfortunately, taking the square root and factoring only work in certain situations.
For example, let’s look at the following quadratic equation:
$x^2 + 6x = -2$
Solving a quadratic equation by taking the square root involves taking the square root of each side of the equation. Because this equation contains a non-squared $\bi x$ (in $\bo6\bi x$), that technique won’t work.
Factoring, on the other hand, involves breaking the quadratic equation into two linear equations that are both equal to zero. Unfortunately, trying to factor this equation doesn’t result in two linear equations!
Both the quadratic formula and completing the square will let you solve any quadratic equation. (In this post, we’re specifically focusing on completing the square.) When you complete the square, you change the equation so that the left side of the equation is a perfect square trinomial. That’s just a fancy way of saying that completing the square is a technique that transforms your quadratic equation from an equation that can’t be factored into one that can.
Completing the square applies to even the trickiest quadratic equations, which you’ll see as we work through the example below.
Your Step-By-Step Guide for How to Complete the Square
Now that we’ve determined that our formula can only be solved by completing the square, let’s look at our example formula again:
$x^2 + 6x = -2$
Step 1: Figure Out What’s Missing
When you look at the equation above, you can see that it doesn’t quite fit the quadratic equation format ($ax^2 + bx + c = 0$). The number that should go in the $c$ spot, which is also known as the constant, is missing. So from a logical perspective, the equation actually looks like this:
$x^2 + 6x +$ __?__ $= -2$
In order to solve this equation, we first need to figure out what number goes into the blank to make the left side of the equation a perfect square. (This missing number is called the constant.) By doing that, we’ll be able to factor the equation like normal.
Step 2: Use the Completing the Square Formula
But at this point, we have no idea what number needs to go in that blank. In order to figure that out, we need to apply the completing the square formula, which is:
$x^2 + 2ax + a^2$
In this case, the $a$ in this equation is the constant, or the number that needs to go in the blank in our quadratic formula above.
Step 3: Apply the Completing the Square Formula to Find the Constant
As long as the coefficient, or number, in front of the $\bi x^\bo2$ is 1, you can quickly and easily use the completing the square formula to solve for $\bi a$.
To do this, you take the middle number, also known as the linear coefficient, and set it equal to $2ax$. Here’s what that would look like for our sample formula:
$6x = 2ax$
This equation is basically asking what number (this is $\bi a$) multiplied by 2 will give us 6.
Now that you know your equation, solving for $a$ is simple: divide each side of the equation by $2x$! So let’s see what that looks like:
$$6x = 2ax$$
Divide each side by $\bo2x$:
$${6x}/{2x} = {2ax}/{2x}$$
Result: $3 = a$
Look at that! We now know that $\bi a =\bo3$!
But we’re not quite done with the completing the square formula yet. In order to determine what the missing constant is, we need to plug our solution for $a$ back into the completing the square formula ($x^2 + 2ax + a^2$). Whatever the result is for $\bi a^\bo2$ is the constant that we’ll plug back into our first equation ($x^2+ 6x +$ __?__ $= -2$). So let’s take a look:
$x^2+ 2ax + a^2$ where $a = 3$
Add $\bi a$ into the equation: $x^2 + 2(3)x + 3^2$
Put in simplest terms: $x^2 + 6x + 9$
So now we know that our constant is 9.
Now it's time to plug in some numbers!
Step 4: Plug the Constant Into the Original Formula
Now that you know the constant, it’s time to put it into the blank in our original formula. Once you do that, the equation will look like this:
Original formula: $x^2 + 6x +$ __?__ $= -2$
Formula with constant: $x^2 + 6x + 9 = -2 + 9$
Put in simplest terms: $x^2+ 6x + 9 = 7$
You might be wondering why we’re adding 9 to the right side of the equation. Well, remember: in math, you can never do something to one side of an equation without doing it to the other side, too. So because we’re adding 9 to our equation to make it a perfect square, we also have to add 9 to the right side of the equation to keep things balanced.
If you forget to add the new constant to the right side of the equation, you won’t get the right answer!
Step 5: Factor the Equation
We’ve already done a lot of work, and there’s still a little more to go. Now it’s time for us to solve the quadratic equation by figuring out what x could be. But now that we’ve turned the left side of our equation into a perfect square, all we have to do is factor like normal.
Completed quadratic formula: $x^2 + 6x + 9 = 7$
Factor left side of the equation: $(x + 3)^2 = 7$
Take the square root: $√{(x + 3)^2}= √7$
Subtract 3: $x = ±√7 - 3$
Final solutions: $x =√{7} - 3$ and $x =√{-7} - 3$
What If There’s a Coefficient in Front of $x^2$?
The step-by-step guide we gave you above only works if there’s no coefficient, or number, in front of $x^2$. If there is a coefficient, you have to eliminate it. Once you do that, you can solve the quadratic equation through the method we outlined above.
So how do you remove the coefficient? Actually, it’s not as hard as it sounds.
To show you how, let’s look at a new quadratic equation:
$2x^2- 12x = -8$
How to Factor Out the 2
n order to remove the 2, you’ll need to divide both sides of the equation by 2. It’s really that simple! So let’s take a look at how that works:
Divide everything by 2: $x^2- 6x = -4$
By doing this, you’ve made the coefficient in front of the $x^2$ into 1, so now you can solve the equation by completing the square like we did above.
Additional Completing the Square Resources
We know that completing the square can be tricky, which is why we’ve compiled a list of resources to help you if you’re still having trouble with how to complete the square.
More Sample Problems
As you already know, practice makes perfect. That’s why it’s important to work as many quadratic equations as you need to in order to feel comfortable solving these types of problems. Luckily for you, completing the square can be used to solve any quadratic equation, so as long as the practice questions are quadratics, you can use them!
One great resource for this is Lamar University’s quadratic equation page, which has a variety of sample problems as well as answers. Another good resource for quadratic equation practice is Math Is Fun’s webpage. If you scroll to the bottom, they have quadratic equation practice questions broken up into categories by difficulty.
Completing the Square Tutorial Videos
If you’re a visual learner, you might find it easier to watch someone solve quadratic equations instead. Khan Academy has an excellent video series on solving quadratic equations, including one video dedicated to showing you how to complete the square. YouTube also has some great resources, including this video on completing the square and this video that shows you how to tackle more advanced quadratic equations.
Completing the Square Calculator
If you want to check your work, there are some completing the square calculators available online. It can be a good way to make sure you’re working problems correctly if you don’t have an answer guide. But be forewarned: relying on a tool like this won’t help you retain the information! Make sure you’re putting in the hard work to learn how to complete the square so you aren’t blindsided by these types of questions on test day.
Now What?
Working with quadratic equations is just one element of algebra you’ll need to master before taking the SAT and ACT. A good place to start is mastering systems of equations, which will help you brush up on your fundamental algebra skills, too.
One of the most helpful math study tools is a chart of useful mathematical equations. Luckily for you, we have a master list of the 31 formulas you must know to conquer the ACT.
If you think you need a more comprehensive study tool, test prep books are one way to go. Here’s a list of our favorite SAT Math prep books that will help set you on the path to success.
