Factoring Quadratic Expressions: A Step-by-Step Guide

Factoring quadratic expressions is a fundamental skill in algebra, essential for solving various mathematical problems. One common method of factoring is performing middle term factorization, which can be applied to expressions like 2a^2 5ab 2b^2. In this article, we will walk through the process of factoring such expressions in a clear, methodical manner.

Understanding the Quadratic Expression

Consider the quadratic expression 2a^2 5ab 2b^2. This expression is a polynomial of degree 2, meaning it involves variables raised to the second power. To factor this expression, we can use the technique of middle term factorization.

Step 1: Identify the Coefficients

The first step in our process is to identify the coefficients of the quadratic expression. For 2a^2 5ab 2b^2, these are as follows:

a 2 b 5 c 2

Here, a is the coefficient of a^2, b is the coefficient of ab, and c is the coefficient of b^2.

Step 2: Calculate the Product of a and c

The next step is to find the product of a and c. In this case:

ac 2*2 4

Step 3: Find Two Numbers that Multiply to ac and Add to b

Look for two numbers that multiply to 4 (the product calculated above) and add to 5 (the coefficient of ab). The numbers 4 and 1 satisfy this condition:

4*14 4 15

Step 4: Rewrite the Middle Term

Substitute the middle term 5ab with 4ab 1ab:

2a^2 5ab 2b^2 2a^2 4ab 1ab 2b^2

Step 5: Group the Terms and Factor Out Common Factors

Group the terms in such a way that common factors can be factored out:

2a^2 4ab 1ab 2b^2 (2a^2 4ab) (1ab 2b^2)

Factor out the common factors in each group:

2a(a 2b) b(a 2b)

Step 6: Factor Out the Common Binomial

Notice that both terms have a common binomial factor. Factor out the common binomial:

(2a b)(a 2b)

Thus, the factorization of 2a^2 5ab 2b^2 is (2a b)(a 2b).

Conclusion

By following these steps, you can factorize quadratic expressions effectively. Middle term factorization is a versatile technique that can be applied to various expressions, making it a valuable skill for solving algebraic problems. Practice with different expressions to improve your proficiency in this fundamental algebraic skill.

Additional Tips

Here are some additional tips to help you factorize quadratic expressions more efficiently:

Identify the Coefficients Promptly: Quickly recognize the coefficients of a^2, ab, and b^2. Use a Systematic Approach: Follow a step-by-step method to ensure accuracy and consistency. Practice Regularly: The more practice you have, the more intuitive the process will become.