How to Split the Middle Term

Oct 13, 2025 by miodeya.com 26 views

Introduction

In algebra, factoring a quadratic expression is a crucial step in solving equations and inequalities. One of the methods used to factor a quadratic expression is by splitting the middle term. This method involves expressing the middle term as a sum or difference of two terms, which can then be factored using the distributive property. In this article, we will learn how to factor the quadratic expression $$2x^2 - 17x + 8$$ by splitting the middle term.

What is Splitting the Middle Term?

Splitting the middle term is a factoring method used to factor a quadratic expression of the form $$ax^2 + bx + c$$. The method involves expressing the middle term $$bx$$ as a sum or difference of two terms, which can then be factored using the distributive property. The general form of a quadratic expression that can be factored by splitting the middle term is:

ax^2 + bx + c = a(x + p)(x + q)

where $$p$$ and $$q$$ are constants.

How to Split the Middle Term

To split the middle term, we need to find two numbers whose product is equal to the product of the coefficient of the middle term and the coefficient of the constant term, and whose sum is equal to the coefficient of the middle term. In the case of the quadratic expression $$2x^2 - 17x + 8$$, we need to find two numbers whose product is equal to $$2 \times 8 = 16$$ and whose sum is equal to $$-17$$.

Step 1: Find the Factors of the Product

The product of the coefficient of the middle term and the coefficient of the constant term is $$2 \times 8 = 16$$. We need to find the factors of 16, which are:

1 \times 16

2 \times 8

4 \times 4

Step 2: Find the Sum of the Factors

We need to find the sum of the factors that we found in Step 1. The sum of the factors is:

1 + 16 = 17

2 + 8 = 10

4 + 4 = 8

Step 3: Choose the Correct Factors

We need to choose the factors that have a sum equal to the coefficient of the middle term, which is $$-17$$. The only pair of factors that have a sum equal to $$-17$$ is $$-8$$ and $$-9$$.

Step 4: Factor the Quadratic Expression

Now that we have found the correct factors, we can factor the quadratic expression $$2x^2 - 17x + 8$$ by splitting the middle term. We can write the quadratic expression as:

2x^2 - 17x + 8 = 2x^2 - 8x - 9x + 8

We can then factor the quadratic expression as:

2x^2 - 17x + 8 = 2x(x - 4) - 1(9x - 8)

We can then factor the quadratic expression further as:

2x^2 - 17x + 8 = (2x - 1)(x - 4)

Conclusion

In this article, we learned how to factor the quadratic expression $$2x^2 - 17x + 8$$ by splitting the middle term. We found the factors of the product of the coefficient of the middle term and the coefficient of the constant term, and then chose the factors that have a sum equal to the coefficient of the middle term. We then factored the quadratic expression by splitting the middle term and obtained the final factorization as $$(2x - 1)(x - 4)$$. This method can be used to factor any quadratic expression of the form $$ax^2 + bx + c$$.

Example Problems

Example 1

Factor the quadratic expression $$x^2 + 5x + 6$$ by splitting the middle term.

Solution

We need to find two numbers whose product is equal to $$1 \times 6 = 6$$ and whose sum is equal to $$5$$. The factors of 6 are:

1 \times 6

2 \times 3

The sum of the factors is:

1 + 6 = 7

2 + 3 = 5

We can then factor the quadratic expression as:

x^2 + 5x + 6 = x^2 + 2x + 3x + 6

We can then factor the quadratic expression further as:

x^2 + 5x + 6 = x(x + 2) + 3(x + 2)

We can then factor the quadratic expression further as:

x^2 + 5x + 6 = (x + 3)(x + 2)

Example 2

Factor the quadratic expression $$3x^2 - 11x + 6$$ by splitting the middle term.

Solution

We need to find two numbers whose product is equal to $$3 \times 6 = 18$$ and whose sum is equal to $$-11$$. The factors of 18 are:

1 \times 18

2 \times 9

3 \times 6

The sum of the factors is:

1 + 18 = 19

2 + 9 = 11

3 + 6 = 9

We can then factor the quadratic expression as:

3x^2 - 11x + 6 = 3x^2 - 9x - 2x + 6

We can then factor the quadratic expression further as:

3x^2 - 11x + 6 = 3x(x - 3) - 2(x - 3)

We can then factor the quadratic expression further as:

3x^2 - 11x + 6 = (3x - 2)(x - 3)

Tips and Tricks

When factoring a quadratic expression by splitting the middle term, make sure to find the factors of the product of the coefficient of the middle term and the coefficient of the constant term.
Make sure to choose the factors that have a sum equal to the coefficient of the middle term.
When factoring a quadratic expression, make sure to factor out the greatest common factor (GCF) of the terms.
When factoring a quadratic expression, make sure to check if the expression can be factored using other methods, such as the quadratic formula.

Conclusion

Introduction

In our previous article, we learned how to factor the quadratic expression $$2x^2 - 17x + 8$$ by splitting the middle term. In this article, we will answer some of the most frequently asked questions about factoring quadratic expressions by splitting the middle term.

Q: What is the purpose of factoring a quadratic expression?

A: The purpose of factoring a quadratic expression is to express it as a product of two or more binomials. This can help us to solve equations and inequalities, and to simplify complex expressions.

Q: How do I know if a quadratic expression can be factored by splitting the middle term?

A: A quadratic expression can be factored by splitting the middle term if it can be written in the form $$ax^2 + bx + c$$, where $$a$$, $$b$$, and $$c$$ are constants. If the product of the coefficient of the middle term and the coefficient of the constant term is a perfect square, then the quadratic expression can be factored by splitting the middle term.

Q: What are the steps to factor a quadratic expression by splitting the middle term?

A: The steps to factor a quadratic expression by splitting the middle term are:

Find the factors of the product of the coefficient of the middle term and the coefficient of the constant term.
Choose the factors that have a sum equal to the coefficient of the middle term.
Factor the quadratic expression by splitting the middle term.

Q: How do I find the factors of the product of the coefficient of the middle term and the coefficient of the constant term?

A: To find the factors of the product of the coefficient of the middle term and the coefficient of the constant term, you can use the following steps:

Multiply the coefficient of the middle term and the coefficient of the constant term.
Find the factors of the product.
List the factors in pairs.

Q: How do I choose the correct factors?

A: To choose the correct factors, you need to find the pair of factors that have a sum equal to the coefficient of the middle term.

Q: What if I get stuck while factoring a quadratic expression by splitting the middle term?

A: If you get stuck while factoring a quadratic expression by splitting the middle term, you can try the following:

Check your work to make sure that you have not made any mistakes.
Try factoring the quadratic expression using a different method, such as the quadratic formula.
Ask for help from a teacher or tutor.

Q: Can I factor a quadratic expression by splitting the middle term if it has a negative coefficient?

A: Yes, you can factor a quadratic expression by splitting the middle term if it has a negative coefficient. However, you need to be careful when choosing the correct factors.

Q: Can I factor a quadratic expression by splitting the middle term if it has a fraction coefficient?

A: Yes, you can factor a quadratic expression by splitting the middle term if it has a fraction coefficient. However, you need to be careful when choosing the correct factors.

Q: What are some common mistakes to avoid when factoring a quadratic expression by splitting the middle term?

A: Some common mistakes to avoid when factoring a quadratic expression by splitting the middle term include:

Not finding the correct factors.
Not choosing the correct factors.
Not factoring the quadratic expression correctly.

Conclusion

In this article, we answered some of the most frequently asked questions about factoring quadratic expressions by splitting the middle term. We hope that this article has been helpful in clarifying any confusion you may have had about this topic. If you have any further questions, please don't hesitate to ask.

Example Problems

Example 1

Factor the quadratic expression $$x^2 + 5x + 6$$ by splitting the middle term.

Solution

We need to find two numbers whose product is equal to $$1 \times 6 = 6$$ and whose sum is equal to $$5$$. The factors of 6 are:

1 \times 6

2 \times 3

The sum of the factors is:

1 + 6 = 7

2 + 3 = 5

We can then factor the quadratic expression as:

x^2 + 5x + 6 = x^2 + 2x + 3x + 6

We can then factor the quadratic expression further as:

x^2 + 5x + 6 = x(x + 2) + 3(x + 2)

We can then factor the quadratic expression further as:

x^2 + 5x + 6 = (x + 3)(x + 2)

Example 2

Factor the quadratic expression $$3x^2 - 11x + 6$$ by splitting the middle term.

Solution

We need to find two numbers whose product is equal to $$3 \times 6 = 18$$ and whose sum is equal to $$-11$$. The factors of 18 are:

1 \times 18

2 \times 9

3 \times 6

The sum of the factors is:

1 + 18 = 19

2 + 9 = 11

3 + 6 = 9

We can then factor the quadratic expression as:

3x^2 - 11x + 6 = 3x^2 - 9x - 2x + 6

We can then factor the quadratic expression further as:

3x^2 - 11x + 6 = 3x(x - 3) - 2(x - 3)

We can then factor the quadratic expression further as:

3x^2 - 11x + 6 = (3x - 2)(x - 3)

Tips and Tricks

When factoring a quadratic expression by splitting the middle term, make sure to find the factors of the product of the coefficient of the middle term and the coefficient of the constant term.
Make sure to choose the factors that have a sum equal to the coefficient of the middle term.
When factoring a quadratic expression, make sure to factor out the greatest common factor (GCF) of the terms.
When factoring a quadratic expression, make sure to check if the expression can be factored using other methods, such as the quadratic formula.