Quadratic equations are fundamental to algebra, and factoring them is a crucial skill. This guide provides a clear, step-by-step approach to mastering this technique, along with examples to solidify your understanding. We'll cover different methods, helping you choose the best approach for various types of quadratic equations.
Understanding Quadratic Equations
Before diving into factoring, let's define what a quadratic equation is. A quadratic equation is a second-degree polynomial equation of the form:
ax² + bx + c = 0
where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. Factoring this equation means rewriting it as a product of two simpler expressions.
Method 1: Factoring when a = 1
This is the simplest case. When the coefficient of x² (a) is 1, we look for two numbers that add up to 'b' and multiply to 'c'.
Example:
Factor x² + 5x + 6 = 0
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Find factors of 'c' (6) that add up to 'b' (5): The numbers 2 and 3 satisfy this condition (2 + 3 = 5 and 2 * 3 = 6).
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Rewrite the equation: (x + 2)(x + 3) = 0
Therefore, the factored form of x² + 5x + 6 is (x + 2)(x + 3).
Method 2: Factoring when a ≠ 1
When 'a' is not equal to 1, the process is slightly more complex. We'll use the AC method:
Example:
Factor 2x² + 7x + 3 = 0
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Find the product of 'a' and 'c': 2 * 3 = 6
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Find factors of 'ac' (6) that add up to 'b' (7): The numbers 6 and 1 satisfy this (6 + 1 = 7 and 6 * 1 = 6).
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Rewrite the equation, splitting the middle term: 2x² + 6x + x + 3 = 0
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Factor by grouping: 2x(x + 3) + 1(x + 3) = 0
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Factor out the common binomial: (2x + 1)(x + 3) = 0
Therefore, the factored form of 2x² + 7x + 3 is (2x + 1)(x + 3).
Method 3: Difference of Squares
This method applies to quadratic equations in the form:
a² - b² = (a + b)(a - b)
Example:
Factor x² - 9 = 0
This is a difference of squares where a = x and b = 3. Therefore:
(x + 3)(x - 3) = 0
Method 4: Perfect Square Trinomial
A perfect square trinomial is a quadratic that can be factored into the square of a binomial. It has the form:
a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²
Example:
Factor x² + 6x + 9 = 0
This is a perfect square trinomial where a = x and b = 3. Therefore:
(x + 3)² = 0
Solving Quadratic Equations After Factoring
Once you've factored the quadratic equation, you can solve for 'x' by setting each factor equal to zero and solving for x.
Example: Using the factored equation (x + 2)(x + 3) = 0 from Method 1:
- x + 2 = 0 => x = -2
- x + 3 = 0 => x = -3
Therefore, the solutions to the equation x² + 5x + 6 = 0 are x = -2 and x = -3.
Practice Makes Perfect!
Mastering quadratic factoring requires practice. Work through numerous examples, trying different methods, and you'll quickly build your skills and confidence. Remember to always check your answers by expanding your factored form to ensure it matches the original equation. With consistent effort, you’ll become proficient in factoring quadratic equations.
