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Factoring quadratics in any form

Tie together everything you learned about quadratic factorization in order to factor various quadratic expressions of any form.

What you need to know for this lesson

The following factoring methods will be used in this lesson:

What you will learn in this lesson

In this article, you will practice putting these methods together to completely factor quadratic expressions of any form.

Intro: Review of factorization methods

MethodExampleWhen is it applicable?
Factoring out common factors= 6x2+3x=3x(2x+1)If each term in the polynomial shares a common factor.
The sum-product pattern= x2+7x+12=(x+3)(x+4)If the polynomial is of the form x2+bx+c and there are factors of c that add up to b.
The grouping method= 2x2+7x+3=2x2+6x+1x+3=2x(x+3)+1(x+3)=(x+3)(2x+1)If the polynomial is of the form ax2+bx+c and there are factors of ac that add up to b.
Perfect square trinomials= x2+10x+25=(x+5)2If the first and last terms are perfect squares and the middle term is twice the product of their square roots.
Difference of squares=  x29=(x3)(x+3)If the expression represents a difference of squares.

Putting it all together

In practice, you'll rarely be told what type of factoring method(s) to use when encountering a problem. So it's important that you develop some sort of checklist to use to help make the factoring process easier.
Here's one example of such a checklist, in which a series of questions is asked in order to determine how to factor the quadratic polynomial.

Factoring quadratic expressions

Before starting any factoring problem, it is helpful to write your expression in standard form.
Once this is the case, you can proceed to the following list of questions:
Question 1: Is there a common factor?
If no, move onto Question 2. If yes, factor out the GCF and continue to Question 2.
Factoring out the GCF is a very important step in the factoring process, as it makes the numbers smaller. This, in turn, makes it easier to recognize patterns!
Question 2: Is there a difference of squares (i.e. x216 or 25x29)?
If a difference of squares pattern occurs, factor using the pattern a2b2=(a+b)(ab). If not, move on to Question 3.
Question 3: Is there a perfect square trinomial (i.e. x210x+25 or 4x2+12x+9)?
If a perfect square trinomial is present, factor using the pattern a2±2ab+b2=(a±b)2. If not, move on to Question 4.
Question 4:
a.) Is there an expression of the form x2+bx+c?
If no, move on to Question 5. If yes, move on to b).
b.) Are there factors of c that sum to b?
If yes, then factor using the sum-product pattern. Otherwise, the quadratic expression cannot be factored further.
Question 5: Are there factors of ac that add up to b?
If you've gotten this far, the quadratic expression must be of the form ax2+bx+c where a1. If there are factors of ac that add up to b, factor using the grouping method. If not, the quadratic expression cannot be factored further.
Following this checklist will help to ensure that you've factored the quadratic completely!
With this in mind, let's try a few examples.

Example 1: Factoring 5x280

Notice that the expression is already in standard form. We can proceed to the checklist.
Question 1: Is there a common factor?
Yes. The GCF of 5x2 and 80 is 5. We can factor this out as follows:
5x280=5(x216)
Question 2: Is there a difference of squares?
Yes. x216=(x)2(4)2. We can use the difference of squares pattern to continue factoring the polynomial as shown below.
5x280=5((x)2(4)2)=5(x+4)(x4)
There are no more quadratics in the expression. We have completely factored the polynomial.
In conclusion, 5x280=5(x+4)(x4).

Example 2: Factoring 4x2+12x+9

The quadratic expression is again in standard form. Let's start the checklist!
Question 1: Is there a common factor?
No. The terms 4x2, 12x and 9 do not share a common factor. Next question.
Question 2: Is there a difference of squares?
No. There’s an x-term so this cannot be a difference of squares. Next question.
Question 3: Is there a perfect square trinomial?
Yes. The first term is a perfect square since 4x2=(2x)2, and the last term is a perfect square since 9=(3)2. Also, the middle term is twice the product of the numbers that are squared since 12x=2(2x)(3).
We can use the perfect square trinomial pattern to factor the quadratic.
=4x2+12x+9=(2x)2+2(2x)(3)+(3)2=(2x+3)2
In conclusion, 4x2+12x+9=(2x+3)2.

Example 3: Factoring 12x63+3x2

This quadratic expression is not currently in standard form. We can rewrite it as 3x2+12x63 and then proceed through the checklist.
Question 1: Is there a common factor?
Yes. The GCF of 3x2, 12x and 63 is 3. We can factor this out as follows:
3x2+12x63=3(x2+4x21)
Question 2: Is there a difference of squares?
No. Next question.
Question 3: Is there a perfect square trinomial?
No. Notice that 21 is not a perfect square, so this cannot be a perfect square trinomial. Next question.
Question 4a: Is there an expression of the form x2+bx+c?
Yes. The resulting quadratic, x2+4x21, is of this form.
Question 4b: Are there factors of c that add up to b?
Yes. Specifically, there are factors of 21 that add up to 4.
Since 7(3)=21 and 7+(3)=4, we can continue to factor as follows:
3(x2+4x21)=3(x2+4x21)=3(x+7)(x3)
In conclusion, 3x2+12x63=3(x+7)(x3).

Example 4: Factoring 4x2+18x10

Notice that this quadratic expression is already in standard form.
Question 1: Is there a common factor?
Yes. The GCF of 4x2, 18x and 10 is 2. We can factor this out as follows:
4x2+18x10=2(2x2+9x5)
Question 2: Is there a difference of squares?
No. Next question.
Question 3: Is there a perfect square trinomial?
No. Next question.
Question 4a: Is there an expression of the form x2+bx+c?
No. The leading coefficient on the quadratic factor is 2. Next question.
Question 5: Are there factors of ac that add up to b?
The resulting quadratic expression is 2x2+9x5, and so we want to find factors of 2(5)=10 that add up to 9.
Since (1)10=10 and (1)+10=9, the answer is yes.
We can now write the middle term as 1x+10x and use grouping to factor:
= 2(2x2+9x5)=2(2x21x+10x5)Split up middle term=2((2x21x)+(10x5))Group terms=2(x(2x1)+5(2x1))Factor out GCFs=2(2x1)(x+5)Factor out 2x1

Check your understanding

1) Factor 2x2+4x16 completely.
صرف 1 جواب چنو

2) Factor 3x260x+300 completely.

3) Factor 72x22 completely.

4) Factor 5x2+5x+15 completely.
صرف 1 جواب چنو

5) Factor 8x212x8 completely.

6) Factor 5618x+x2 completely.

7) Factor 3x2+27 completely.
صرف 1 جواب چنو