Factoring Using the AC Method for Quadratics
A quadratic is a polynomial in the form
where a, b, and c are real numbers.
When you have a quadratic expression, you can factor it using the AC Method.
Here are some steps to follow when factoring using the AC Method.
When you have a quadratic expression, you can factor it using the AC Method.
Here are some steps to follow when factoring using the AC Method.
- Check for a GCF - if there is one, factor like you did in the GCF lesson, if there is not, move to step 2
- Multiply a by c and write it in the first column (factors) of the t-chart shown below. Write b in the second column (sum).
- Find factors of ac that have a sum of b.
- Make a box and fill in the terms shown below. Don't forget to put your variable with your factors! It does not matter which factor goes in which box!
- Find the GCF of each column and write it above the column. Find the GCF of each row and write it to the right of the row. Keep the sign of the first term you see as you read down or left to right!
Remember, if there was a GCF, you should include it in your final answer!
Again, you can check that you factored correctly by using FOIL.
Here is an example:
Again, you can check that you factored correctly by using FOIL.
Here is an example:
Step 1: Check for a GCF.
There is not a greatest common factor (other than 1), so we move on to step 2.
Step 2: Multiply a by c and write it in the factors column of your table. Write b in the sum column.
Since a = 3 and c = -20 (make sure you keep track of your signs) ac = -60, b = -11
There is not a greatest common factor (other than 1), so we move on to step 2.
Step 2: Multiply a by c and write it in the factors column of your table. Write b in the sum column.
Since a = 3 and c = -20 (make sure you keep track of your signs) ac = -60, b = -11
Step 3: Find factors of -60 that have a sum of -11.
If you cannot find them quickly, start writing all of your factor pairs starting with 1. If the sum is correct, you have found your factors. If it is not correct, cross it off and try the next pair. This is demonstrated in the below.
If you cannot find them quickly, start writing all of your factor pairs starting with 1. If the sum is correct, you have found your factors. If it is not correct, cross it off and try the next pair. This is demonstrated in the below.
Step 4: Make a box and fill in the terms, remembering to put the variable with your factors.
Remember the first term, or squared term, goes in the top left corner and the last term, or constant, goes in the bottom right. The factors go across the diagonal - it does not matter which factor goes in which box!
Remember the first term, or squared term, goes in the top left corner and the last term, or constant, goes in the bottom right. The factors go across the diagonal - it does not matter which factor goes in which box!
Step 5: Find the GCF of each column and write it above the column. Find the GCF of each row and write it to the right of the row.
Remember to keep the sign of the first term you see as you read down or left to right!
Remember to keep the sign of the first term you see as you read down or left to right!
Circle the GCF you found. Rewrite them in your final factored form as shown below.
Remember that you can check that you factored correctly by using FOIL.
Here is another example:
Step 1: Check for a GCF.
The greatest common factor is 2, so factoring out the GCF, we get
The greatest common factor is 2, so factoring out the GCF, we get
Step 2: Multiply a by c and write it in the factors column of your table. Write b in the sum column.
Since a = 6 and c = 7 (make sure you keep track of your signs) ac = 42, b = -17
Since a = 6 and c = 7 (make sure you keep track of your signs) ac = 42, b = -17
Step 3: Find factors of 42 that have a sum of -17.
If you cannot find them quickly, start writing all of your factor pairs starting with 1. If the sum is correct, you have found your factors. If it is not correct, cross it off and try the next pair. This is demonstrated in the below.
If you cannot find them quickly, start writing all of your factor pairs starting with 1. If the sum is correct, you have found your factors. If it is not correct, cross it off and try the next pair. This is demonstrated in the below.
Step 4: Make a box and fill in the terms, remembering to put the variable with your factors.
Remember the first term, or squared term, goes in the top left corner and the last term, or constant, goes in the bottom right. The factors go across the diagonal - it does not matter which factor goes in which box!
Remember the first term, or squared term, goes in the top left corner and the last term, or constant, goes in the bottom right. The factors go across the diagonal - it does not matter which factor goes in which box!
Step 5: Find the GCF of each column and write it above the column. Find the GCF of each row and write it to the right of the row.
Remember to keep the sign of the first term you see as you read down or left to right!
Remember to keep the sign of the first term you see as you read down or left to right!
Circle the GCF you found. Rewrite them in your final factored form as shown below. Don't forget to bring down the GCF you factored out in step 1 to your final answer!
Remember to check that you factored correctly by using FOIL and the distributive property.
Watch the following videos to learn more about factoring quadratics.
Watch the following videos to learn more about factoring quadratics.
Box Factoring Example 1 by 1bshirley: https://www.youtube.com/watch?v=uJCZlbzX6u0
BOX METHOD of Factoring Polynomials.m4v by tube4horst: https://www.youtube.com/watch?v=_Wb_CT-1VN8
Practice:
Use the following website to practice factoring quadratics:
https://www.ixl.com/math/algebra-2/factor-quadratics
https://www.ixl.com/math/algebra-2/factor-quadratics
Quick Check:
Complete the following quick check. When you are finished (and have mastered) the AC Method, move onto "Post-Test."
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