Complete Factorization of Quadratic Expressions: A Guide with Practical Examples

Mastering the art of factorization is a fundamental skill in algebra. This article will walk you through the step-by-step process of factoring a quadratic expression, specifically applying it to the expression y2 - y - 30.

Understanding the Basics

When faced with a quadratic expression like y2 - y - 30, the goal is to express it as a product of its factors, typically in the form of linear expressions: (y a)(y b).

Step-by-Step Factorization

Let's factorize y2 - y - 30 using a systematic approach:

Identify the product and sum required: For a quadratic expression ay2 by c, the product is c and the sum is b. In our case, a 1, b -1, and c -30. Find two numbers whose sum is -1 and whose product is -30. Check the factors: The two numbers that satisfy these conditions are 5 and -6 because 5 (-6) -1 and 5 * (-6) -30. Rewrite the middle term using these numbers: y2 - y - 30 y2 5y - 6y - 30. Group the terms to factor by grouping: (y2 5y) - (6y 30). Factor out the common factors from each group: y(y 5) - 6(y 5). Factor out the common binomial: (y 5)(y - 6).

Conclusion and Verification

Therefore, the complete factorization of y2 - y - 30 is: (y 5)(y - 6).

Further Exploration

Understanding this process can help solve quadratic equations. For instance, if you set y2 - y - 30 0, the solutions are given by the factorization method:

(y 5)(y - 6) 0 implies y -5 or y 6.

These steps are crucial for students and professionals working with algebra, providing a solid foundation for more advanced mathematical operations and problem-solving techniques.