Factoring a perfect square trinomial involves recognizing its unique pattern and applying a specific formula. Here's a step-by-step guide:
1. Identify a Perfect Square Trinomial
A perfect square trinomial follows one of these two patterns:
- a² + 2ab + b²
- a² - 2ab + b²
To determine if a trinomial is a perfect square:
- Check if the first and last terms are perfect squares. This means you can take the square root of both terms and get a whole number or simple expression.
- Verify that the middle term is twice the product of the square roots of the first and last terms. In other words, is the middle term equal to 2 (√first term) (√last term)?
2. Apply the Appropriate Formula
Once you've confirmed it's a perfect square trinomial, apply the correct formula:
- If the trinomial is in the form a² + 2ab + b², then it factors to (a + b)²
- If the trinomial is in the form a² - 2ab + b², then it factors to (a - b)²
3. Determine 'a' and 'b'
- Find 'a' by taking the square root of the first term.
- Find 'b' by taking the square root of the last term.
4. Substitute 'a' and 'b' into the Formula
Substitute the values you found for 'a' and 'b' into the appropriate formula from step 2.
5. Write the Factored Form
Write the expression in the factored form: (a + b)² or (a - b)².
Example 1: Factoring x² + 6x + 9
- Identify: The first term (x²) and last term (9) are perfect squares.
- Verify: √x² = x and √9 = 3. Is 2 x 3 = 6x? Yes, it is.
- Formula: Since it's in the form a² + 2ab + b², use (a + b)²
- Determine a and b: a = √x² = x, b = √9 = 3
- Substitute: (x + 3)²
- Factored Form: (x + 3)²
Example 2: Factoring 4x² - 20x + 25
- Identify: The first term (4x²) and last term (25) are perfect squares.
- Verify: √4x² = 2x and √25 = 5. Is 2 2x 5 = 20x? Yes, it is.
- Formula: Since it's in the form a² - 2ab + b², use (a - b)²
- Determine a and b: a = √4x² = 2x, b = √25 = 5
- Substitute: (2x - 5)²
- Factored Form: (2x - 5)²
