So you're tackling quadratic expressions and want to master factorising? Let's break down how to factorise y² + 7y + 12, and then explore broader strategies to conquer any quadratic equation. This guide will provide tried-and-tested tips to improve your skills and boost your confidence in algebra.
Understanding Factorisation
Before diving into the specifics of y² + 7y + 12, let's clarify what factorisation means. Essentially, it's the process of breaking down a mathematical expression into simpler parts that, when multiplied together, give you the original expression. Think of it like reverse multiplication. For quadratic expressions like y² + 7y + 12, we're looking for two binomial expressions (expressions with two terms) that, when multiplied, result in the original quadratic.
Step-by-Step Factorisation of y² + 7y + 12
Here's a clear, step-by-step approach:
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Identify the Coefficients: Our quadratic is in the form ay² + by + c, where a = 1, b = 7, and c = 12.
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Find Two Numbers: We need to find two numbers that:
- Add up to 'b' (7): This is the coefficient of the 'y' term.
- Multiply to 'c' (12): This is the constant term.
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The Winning Pair: After some brainstorming (or a bit of trial and error if you are just starting out), you'll find that 3 and 4 fit the bill perfectly: 3 + 4 = 7 and 3 x 4 = 12.
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Construct the Factorised Form: Now, we can write the factorised form as: (y + 3)(y + 4)
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Verify Your Answer: To double-check, expand (y + 3)(y + 4) using the FOIL method (First, Outer, Inner, Last). You should arrive back at y² + 7y + 12.
Mastering Factorisation: Tips and Tricks
Here are some additional tips to help you become a factorisation pro:
Practice Regularly: The key to mastering factorisation is consistent practice. Work through various examples, starting with simpler quadratics and gradually increasing the difficulty.
Learn to Recognize Patterns: As you practice, you'll begin to recognize common patterns and shortcuts. This will significantly speed up your factorisation process.
Use the Quadratic Formula (When Needed): For more complex quadratics where finding the two numbers is difficult, the quadratic formula can help you find the roots and subsequently factorise the expression. Remember the quadratic formula: y = [-b ± √(b² - 4ac)] / 2a
Factor Out the Greatest Common Factor (GCF) First: Always check for a greatest common factor before attempting other factorisation methods. This simplifies the expression and often makes it easier to factorise.
Utilize Online Resources: Numerous websites and educational videos offer further explanation and practice problems. Search for "factorising quadratic equations" to find helpful resources.
Beyond y² + 7y + 12: Expanding Your Skills
The techniques outlined above apply to various quadratic expressions. Practice with different coefficients (a, b, and c) to build proficiency. Remember to always check your answer by expanding the factorised form.
By diligently following these tips and practicing regularly, you'll confidently master factorising quadratic expressions like y² + 7y + 12, and tackle even more complex algebraic problems. Good luck!
