How to Solve Quadratic Equations by Completing the Square Formula
What is the Completing the Square Formula and how can you use it to solve problems?
Welcome to this free lesson guide that accompanies this Completing the Square Explained! video tutorial, where you will learn the answers to the following key questions and information:
What is the completing the square formula?
How can I solve by completing the square?
How can I master solving quadratic equations by completing the square?
What are the completing the square steps?
This Complete Guide to the Completing the Square includes several examples, a step-by-step tutorial, an animated video mini-lesson, and a free worksheet and answer key.
*This lesson guide accompanies our animated Completing the Square Explained! YouTube.
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When can you use the completing the square method to solve quadratic equations?
Solving by completing the square is used to solve quadratic equations in the following form:
Note that a quadratic can be rearranged by subtracting the constant, c, from both sides as follows:
Figure 1
These are two different ways of expressing a quadratic.
Keep this in mind while solving the following problems:
Completing the Square Formula
The following method is less of a formula and more like completing the square steps:
Example: Solve the following quadratic by completing the square.
Note that a=1 and b=-6
Notice that a=1, and b=-6, but what about the constant, c?
Completing the Square Step 1 of 3: Rearrange if Possible
To complete the square, you need to have all of the constants (numbers that are not attached to variables) on the right side of the equals sign.
In this example, you can achieve this by subtracting 9 from both sides and simplifying as follows:
Now that you have rearranged the quadratic so that all of the constants are on the right side of the equals sign, you are ready for step 2!
Completing the Square Step 2 of 3: +(b/2)^2 to both sides
The second step to solving by completing the square is to add (b/2)^2 to both sides of the equation.
Remember the alternate way to write a quadratic from Figure 1 earlier on? Let’s look at it again with our current equation directly below it for reference.
Figure 1
Step two requires that you add (b/2)^2 to both sides and it should be clear that, in this example, b equals -6.
So to find the value of (b/2)^2, just plug in -6 for b and solve as follows:
In this case, (b/2)^2 equals 9
Since (b/2)^2 equals 9, go ahead and add 9 to both sides of the equals sign as follows:
You can simplify the right side of the equal sign by adding 16 and 9.
Notice that you can simplify the right side of the equal sign by adding 16 and 9 to get 25.
Now you are ready for the final step!
Completing the Square Step 3 of 3: Factor and Solve
Notice that, on the left side of the equation, you have a trinomial that is easy to factor.
The factors of the trinomial on the left side of the equals sign are (x-3)(x-3) or (x-3)^2
Completing the square will allows leave you with two of the same factors.
Expressing the factors are (x-3)^2 instead of (x-3)(x-3) is very important because it allows you to solve the problem as follows:
Simplify by taking the square root of both sides.
This is what is left after taking the square root of both sides.
After taking the square root of both sides, you are left with x-3 = +/- 5.
Next, to get x by itself, add 3 to both sides as follows.
And to find your solutions, simply perform x = 3 + 5 AND x = 3 - 5 to get your answer as follows:
Answer: x= 8 and x = -2
This method will apply to solving any quadratic equation! Let’s quickly review the completing the square formula method steps below and then take a look at a few more examples.
Solving by Completing the Square Steps Method Review:
Solve by Completing the Square Problems
Example 1:
Solve for x by completing the square.
STEP 1/3: REARRANGE IF NECESSARY
Start by moving all of the constants to the right side of the equals sign as follows:
Leave yourself some room to work with!
STEP 2/3: +(b/2)^2 to both sides
In this example, b=2, so (b/2)^2 = (2/2)^2 = (1)^2 = 1
So, the next step is to add 1 to both sides as follows:
STEP 3/3: Factor and Solve
For the final step, factor the trinomial on the left side of the equals sign and solve for x as follows:
The trinomial factors to (x+1)(x+1) or (x+1)^2
The square root of 8 is approximately 2.83
These are the solutions!
Answer: x=1.83 and x=-3.83
Solve by Completing the Square Examples
Example 2:
Solve for x by completing the square.
On this final example, follow the complete the square formula 3-step method for finding the solutions* as follows:
*Note that this problem will have imaginary solutions.
Step 1/3: Move the constants to the right side.
Step 2/3: Add (b/2)^2 to both sides.
Step 3/3: Factor and Solve
You can get a more detailed step-by-step explanation of how to solve the above example by watching the video tutorial below starting from minute 7:36.
Completing the Square Explained: Video Tutorial
Still confused? Check out the animated video lesson below:
Check out the video lesson below to learn more about the completing the square and to see more completing the square problems solved step-by-step:
Extra Practice: Free Completing the Square Worksheet
Free Worksheet!
Are you looking for some extra practice? Click the links below to download your free worksheets and answer key:
Completing the Square Practice Worksheet:
CLICK HERE TO DOWNLOAD YOUR FREE WORKSHEET
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By Anthony Persico
Anthony is the content crafter and head educator for YouTube's MashUp Math. You can often find me happily developing animated math lessons to share on my YouTube channel . Or spending way too much time at the gym or playing on my phone.