Factoring quadratic expressions like x² + 5x + 6 is a fundamental skill in algebra. Mastering this technique opens doors to solving more complex equations and understanding various mathematical concepts. This guide provides high-quality suggestions to help you learn how to factorize x² + 5x + 6 effectively.
Understanding the Basics of Quadratic Expressions
Before diving into factorization, it's crucial to understand what a quadratic expression is. A quadratic expression is an algebraic expression of the form ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. In our example, x² + 5x + 6, a = 1, b = 5, and c = 6.
The Goal of Factorization
Factorization aims to rewrite the quadratic expression as a product of two simpler expressions (binomials). This process is essentially the reverse of expanding brackets using the distributive property (FOIL method). Our goal is to find two binomials that, when multiplied together, equal x² + 5x + 6.
Step-by-Step Factorization of x² + 5x + 6
Here's a breakdown of how to factorize x² + 5x + 6:
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Identify 'a', 'b', and 'c': As mentioned earlier, in x² + 5x + 6, a = 1, b = 5, and c = 6.
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Find two numbers that add up to 'b' (5) and multiply to 'c' (6): This is the key step. We need to find two numbers that satisfy both conditions. Let's consider the factors of 6:
- 1 and 6 (1 + 6 = 7, 1 * 6 = 6) - This pair doesn't add up to 5.
- 2 and 3 (2 + 3 = 5, 2 * 3 = 6) - This pair works!
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Rewrite the expression using the two numbers: We can now rewrite the expression as: x² + 2x + 3x + 6
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Factor by grouping: Group the terms and factor out common factors:
- (x² + 2x) + (3x + 6)
- x(x + 2) + 3(x + 2)
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Factor out the common binomial: Notice that (x + 2) is common to both terms. Factor it out:
- (x + 2)(x + 3)
Therefore, the factorization of x² + 5x + 6 is (x + 2)(x + 3).
Verification: Expanding the Factored Form
To verify your factorization, expand the factored form (x + 2)(x + 3) using the FOIL method (First, Outer, Inner, Last):
- First: x * x = x²
- Outer: x * 3 = 3x
- Inner: 2 * x = 2x
- Last: 2 * 3 = 6
Combining the terms, we get x² + 3x + 2x + 6 = x² + 5x + 6. This matches the original expression, confirming our factorization is correct.
Tips and Tricks for Successful Factorization
- Practice Regularly: The more you practice, the faster and more confident you'll become.
- Start with Easy Examples: Begin with simpler quadratic expressions before tackling more challenging ones.
- Use Visual Aids: Diagrams and online tools can help visualize the process.
- Check Your Work: Always expand your factored form to verify your answer.
Mastering factorization is a crucial step in your algebra journey. By following these suggestions and practicing diligently, you'll develop a strong understanding of this important mathematical concept and confidently tackle more complex problems. Remember, consistent practice is the key to success!
