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### Course: Алгебра 1 > Unit 13

Lesson 8: Квадраттық теңдеулерді көбейткішке жіктеу жолдары# 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:

#### Бұл сабақта сен нені үйренесің

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

## Intro: Review of factorization methods

Method | Example | When is it applicable? |
---|---|---|

Factoring out common factors | If each term in the polynomial shares a common factor. | |

The sum-product pattern | If the polynomial is of the form | |

The grouping method | If the polynomial is of the form | |

Perfect square trinomials | If the first and last terms are perfect squares and the middle term is twice the product of their square roots. | |

Difference of squares | If the expression represents a difference of squares. |

## Барлығын қорытындылау

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.**${x}^{2}-16$ or $25{x}^{2}-9$ )?

If a difference of squares pattern occurs, factor using the pattern

**Question 3: Is there a perfect square trinomial (i.e.**${x}^{2}-10x+25$ or $4{x}^{2}+12x+9$ )?

If a perfect square trinomial is present, factor using the pattern

**Question 4:**

a.) Is there an expression of the form? ${x}^{2}+bx+c$

If no, move on to Question 5. If yes, move on tob).

b.) Are there factors ofthat sum to $c$ ? $b$

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

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 $5{x}^{2}-80$

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

**Question 2: Is there a difference of squares?**

Yes.

There are no more quadratics in the expression. We have completely factored the polynomial.

In conclusion, $5{x}^{2}-80=5(x+4)(x-4)$ .

## Example 2: Factoring $4{x}^{2}+12x+9$

The quadratic expression is again in standard form. Let's start the checklist!

**Question 1: Is there a common factor?**

No. The terms

**Question 2: Is there a difference of squares?**

No. There’s an

**Question 3: Is there a perfect square trinomial?**

Yes. The first term is a perfect square since

We can use the perfect square trinomial pattern to factor the quadratic.

In conclusion, $4{x}^{2}+12x+9=(2x+3{)}^{2}$ .

## Example 3: Factoring $12x-63+3{x}^{2}$

This quadratic expression is not currently in standard form. We can rewrite it as $3{x}^{2}+12x-63$ and then proceed through the checklist.

**Question 1: Is there a common factor?**

Yes. The GCF of

**Question 2: Is there a difference of squares?**

No. Next question.

**Question 3: Is there a perfect square trinomial?**

No. Notice that

**Question 4a: Is there an expression of the form**${x}^{2}+bx+c$ ?

Yes. The resulting quadratic,

**Question 4b: Are there factors of**$c$ that add up to $b$ ?

Yes. Specifically, there are factors of

Since $7\cdot (-3)=-21$ and $7+(-3)=4$ , we can continue to factor as follows:

In conclusion, $3{x}^{2}+12x-63=3(x+7)(x-3)$ .

## Example 4: Factoring $4{x}^{2}+18x-10$

Notice that this quadratic expression is already in standard form.

**Question 1: Is there a common factor?**

Yes. The GCF of

**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**${x}^{2}+bx+c$ ?

No. The leading coefficient on the quadratic factor is

**Question 5:**

**Are there factors of**$ac$ that add up to $b$ ?

The resulting quadratic expression is

Since $(-1)\cdot 10=-10$ and $(-1)+10=9$ , the answer is yes.

We can now write the middle term as $-1x+10x$ and use grouping to factor:

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