Factoring Quadratic Trinomials: A Step-by-Step Guide to 2a^2 - 7ab 6b^2
When dealing with algebraic expressions, particularly quadratic trinomials like 2a^2 - 7ab 6b^2, it's important to understand the process of factoring. In this guide, we will break down the process step-by-step, providing clear explanations and examples.
Understanding Quadratic Trinomials
A quadratic trinomial is a polynomial with three terms, where the highest power of the variable is two. The expression 2a^2 - 7ab 6b^2 is a quadratic trinomial with terms involving the variables a and b. The goal is to factor this expression into its simplest form, which often involves finding common factors or grouping terms.
Factoring the Trinomial: 2a^2 - 7ab 6b^2
Let's start by factoring the quadratic trinomial (2a^2 - 7ab 6b^2) using a detailed approach:
1. **Identify the Coefficients**: The expression is 2a^2 - 7ab 6b^2. Here, the coefficients are 2, -7, and 6.
2. **Find the Product and Pair of Numbers**: We need to find two numbers that multiply to 2 * 6 12 and add up to -7. The numbers that work are -3 and -4 since (-3) * (-4) 12 and (-3) (-4) -7.
3. **Rewrite the Middle Term**: Using -3ab and -4ab, we can rewrite the middle term -7ab as -3ab - 4ab.
The expression becomes:
2a^2 - 3ab - 4ab 6b^2
4. **Group the Terms**: Now, group the terms in pairs:
(2a^2 - 3ab) (-4ab 6b^2)
5. **Factor by Grouping**: Factor out the common factors from each pair:
a(2a - 3b) - 2b(2a - 3b)
6. **Factor Out the Common Binomial Factor**: The common factor in both terms is (2a - 3b), so we can factor it out:
(2a - 3b)(a - 2b)
The Factored Form of 2a^2 - 7ab 6b^2
The factored form of the expression 2a^2 - 7ab 6b^2 is:
(2a - 3b)(a - 2b)
This is a more compact and useful form of the expression, as it reveals the factors of the quadratic trinomial.
Conclusion
Factoring quadratic trinomials can be a useful tool in algebra, allowing us to solve equations, simplify expressions, and perform various other operations. By understanding the process, you can apply it to a wide range of similar expressions.
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quadratic trinomials factoring algebraic expressionsFurther Reading and Exercises
If you want to dive deeper into this topic, you might want to explore further reading and practice exercises on quadratic trinomials and factoring techniques. Practice is key to mastering this skill.