Factoring quadratic expressions like y² + 2y - 15 is a fundamental skill in algebra. Mastering this technique opens doors to solving more complex equations and tackling advanced mathematical concepts. This guide will walk you through the essential steps and routines to help you confidently factor this and similar expressions.
Understanding Quadratic Expressions
Before diving into the factoring process, let's refresh our understanding of quadratic expressions. A quadratic expression is an expression of the form ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'x' is the variable. In our example, y² + 2y - 15, 'a' = 1, 'b' = 2, and 'c' = -15.
Key Concepts:
- Coefficient: The numerical factor of a term (e.g., 1 in y², 2 in 2y, -15 in -15).
- Constant: A term without a variable (e.g., -15).
- Variable: A symbol representing an unknown value (e.g., y).
Factoring y² + 2y - 15: A Step-by-Step Guide
The goal of factoring is to rewrite the quadratic expression as a product of two simpler expressions (binomials). Here's how to factor y² + 2y - 15:
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Find Factors of 'c' that Add Up to 'b': We need to find two numbers that multiply to -15 (our 'c' value) and add up to 2 (our 'b' value). Let's list the factor pairs of -15:
- 1 and -15 (1 + (-15) = -14)
- -1 and 15 (-1 + 15 = 14)
- 3 and -5 (3 + (-5) = -2)
- -3 and 5 (-3 + 5 = 2)
We found our pair: -3 and 5!
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Rewrite the Expression: Now, rewrite the original expression using the factors we found:
y² + 2y - 15 = y² - 3y + 5y - 15
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Factor by Grouping: Group the terms in pairs:
(y² - 3y) + (5y - 15)
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Factor Out Common Factors: Factor out the greatest common factor (GCF) from each pair:
y(y - 3) + 5(y - 3)
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Factor Out the Common Binomial: Notice that both terms now share the common binomial (y - 3). Factor it out:
(y - 3)(y + 5)
Therefore, the factored form of y² + 2y - 15 is (y - 3)(y + 5).
Practice Makes Perfect
The best way to master factoring quadratic expressions is through consistent practice. Try factoring these expressions using the same steps outlined above:
- x² + 7x + 12
- z² - 4z - 21
- a² + 5a - 6
Troubleshooting Common Mistakes
- Sign Errors: Be meticulous with positive and negative signs. A simple sign error can derail the entire factoring process.
- Incorrect Factor Pairs: Double-check that your chosen factor pairs multiply to 'c' and add up to 'b'.
- Forgetting to Check: Always expand your factored expression to verify it equals the original quadratic expression.
By consistently practicing these routines and understanding the underlying concepts, you'll develop a solid grasp of factoring quadratic expressions, a cornerstone skill for success in algebra and beyond. Remember, the key is persistent practice and careful attention to detail.
