Okay, so now you understand why you want to factor a quadratic expression, but how do you do it? It’s not easy to look at x2 + 3x − 10 and see that it equals (x + 5)(x − 2).
To get started, try solving the following puzzle. (Hint: It involves addition and multiplication.) The first two are done for you:
Have you figured out the trick to this puzzle? The answers are as follows:
The way the diamonds work is that you multiply the two numbers on the sides to obtain the top number, and you add them to arrive at the bottom number.
Take another look at the connection between (x + 2)(x + 3) and x2 + 5x + 6:
The 2 and the 3 play two important roles in building the quadratic expression:
So when you are trying to factor a quadratic expression such as x2 + 5x + 6, the key is to find the two numbers whose product equals the final term (6) and whose sum equals the coefficient of the middle term (the 5 in 5x). In this case, the two numbers that multiply to 6 and add up to 5 are 2 and 3: 2 × 3 = 6 and 2 + 3 = 5.
So the diamond puzzle is just a visual representation of this same goal. For any quadratic expression, take the final term (the constant) and place it in the top portion of the diamond. Take the coefficient of the middle term (in this case, the 5 in 5x) and place it in the lower portion of the diamond. For instance, if the middle term is 5x, take the 5 and place it at the bottom of the diamond. Now go through the entire process with a new example: x2 + 7x + 12.
The final term is 12, and the coefficient of the middle term is 7, so the diamond will look like this:
When factoring quadratics (or solving the diamond puzzle), it is better to focus first on determining which numbers could multiply to the final term. The reason is that these problems typically deal only with integers, and there are far fewer pairs of integers that will multiply to a certain product than will add to a certain sum. For instance, in this problem, there are literally an infinite number of integer pairs that can add to 7 (remember, negative numbers are also integers: −900,000 and 900,007 sum to 7, for instance). On the other hand, there are only a few integer pairs that multiply to 12. You can actually list them all out: 1 and 12, 2 and 6, and 3 and 4. Because 1 and 12 sum to 13, they don’t work; 2 and 6 sum to 8, so they don’t work either. However, 3 and 4 sum to 7, so this pair of numbers is the one you want. So your completed diamond looks like this:
Now, because your numbers are 3 and 4, the factored form of your quadratic expression becomes (x + 3)(x + 4).
Note: if you are factoring x2 + 7x + 12 = 0, you get (x + 3)(x + 4) = 0, so your solutions are negative 3 or negative 4, not 3 and 4 themselves. Remember, if you have (x + 3)(x + 4) = 0, then either x + 3 = 0 or x + 4 = 0.
Here’s another example with one important difference. Solve the diamond puzzle for this quadratic expression: x2 − 9x + 18. Your diamond looks like this:
You need two numbers that multiply to positive 18, but sum to −9. Here, you know the product is positive, but the sum is negative. So when the top number is positive and the bottom number is negative, the two numbers you are looking for will both be negative.
Once again, it will be easier to start by figuring out what pairs of numbers can multiply to 18. In this case, three different pairs all multiply to 18: −1 and −18, −2 and −9, and −3 and −6. The pair −3 and −6, however, is the only pair of numbers that also sums to −9, so this is the pair you want. Fill in the missing numbers, and your diamond becomes:
If the numbers on the left and right of the diamond are −3 and −6, the factored form of the quadratic expression becomes (x − 3)(x − 6), so the solutions are positive 3 or positive 6.
To recap, when the final term of the quadratic is positive, the two numbers you are looking for will either both be positive or both be negative. If the middle term is positive, as in the case of x2 + 7x + 12, the numbers will both be positive (3 and 4). If the middle term is negative, as in the case of x2 − 9x + 18, the numbers will both be negative (−3 and −6).
Factor the following quadratic expressions.
x2 + 14x + 33
x2 − 14x + 45
The previous section dealt with quadratic equations in which the final term was positive. This section discusses how to deal with quadratics in which the final term is negative. The basic method is the same, although there is one important twist.
Take a look at the quadratic expression x2 + 3x − 10. Start by creating your diamond:
You are looking for two numbers that will multiply to −10. The only way for the product of two numbers to be negative is for one of them to be positive and one of them to be negative. That means that in addition to figuring out pairs of numbers that multiply to 10, you also need to worry about which number will be positive and which will be negative. For the moment, disregard the signs. There are only two pairs of integers that multiply to 10: 1 and 10 and 2 and 5. Start testing the pair 1 and 10, and see what you can learn.
Try making 1 positive and 10 negative. If that were the case, the factored form of the expression would be (x + 1)(x − 10). FOIL it and see what it would look like:
The sum of 1 and −10 is −9, but you want 3. That’s not correct, so try reversing the signs. Now see what happens if you make 1 negative and 10 positive. The factored form would now be (x − 1)(x + 10). Once again, FOIL it out:
Again, this doesn’t match your target. The sum of −1 and 10 is not 3. Compare these examples to the examples in the last section. Notice that, with the examples in the last section, the two numbers summed to the coefficient of the middle term (in the example x2 + 7x + 12, the two numbers you wanted, 3 and 4, summed to 7, which is the coefficient of the middle term). In these two examples, however, because one number was positive and one number was negative, it is actually the difference of 1 and 10 that gave us the coefficient of the middle term.
This will be discussed further as the example continues. For now, to factor quadratics in which the final term is negative, you actually ignore the sign initially and look for two numbers that multiply to the coefficient of the final term (ignoring the sign) and whose difference is the coefficient of the middle term (ignoring the sign).
Going back to the example, the pair of numbers 1 and 10 did not work, so look at the pair 2 and 5. Notice that the coefficient of the middle term is 3, and the difference of 2 and 5 is 3. This has to be the correct pair, so all you need to do is determine whether your factored form is (x + 2)(x − 5) or (x − 2)(x + 5). Take some time now to FOIL both expressions and figure out which one is correct.
You should have come to the conclusion that (x − 2)(x + 5) was the correctly factored form of the expression. That means your diamond looks like this:
To recap, the way to factor any quadratic expression where the final term is negative is as follows:
Work through one more example to see how this works. What is the factored form of x2 − 4x − 21? Take some time to work through it for yourself before looking at the explanation.
First, start your diamond. It looks like this:
Because the coefficient of the final term (−21) is negative, you’re going to ignore the signs for the moment, and focus on finding pairs of integers that will multiply to 21. The only possible pairs are 1 and 21, and 3 and 7. Next, take the difference of both pairs: 21 − 1 = 20 and 7 − 3 = 4. The second pair matches the −4 on the bottom of the diamond (because you are ignoring the sign of the −4 at this stage), so 3 and 7 is the correct pair of numbers.
Now all that remains is to determine the sign of each. The coefficient of the middle term (−4) is negative, so you need to assign the negative sign to the greater of the two numbers, 7. That means that the 3 will be positive. Subsequently, the correctly factored form of the quadratic expression is (x + 3)(x − 7):
Factor the following expressions.
x2 + 3x − 18
x2 − 5x − 66