Perfect Square Trinomial
Expands the square of a binomial.
Formula
Variables
Example
(x+3)² = x² + 6x + 9
How it works
A perfect square trinomial appears when you multiply a binomial by itself. Using distributive multiplication, each term in the first bracket multiplies each term in the second, so you get two matching middle terms that combine to make 2ab and two square terms, a b2 and b b2. The sign in the middle follows the sign inside the binomial: plus gives a positive middle term, and minus gives a negative middle term.
When to use it
Use this formula when you need to expand or recognize expressions like (x+5)^2 or (3y-2)^2. It is also useful for factoring trinomials that match the pattern a^2 b1 2ab + b^2.
Worked example, step by step
Expand (x - 4)^2 Step 1: Identify a = x and b = 4 Step 2: Use the formula (a - b)^2 = a^2 - 2ab + b^2 Step 3: Substitute the values: (x - 4)^2 = x^2 - 2(x)(4) + 4^2 Step 4: Simplify: x^2 - 8x + 16 Final answer: (x - 4)^2 = x^2 - 8x + 16
Common mistakes to avoid
- Forgetting to double the product 2ab in the middle term
- Leaving out the sign of the binomial when expanding
- Squaring only one term instead of both terms in the binomial
Did You Know?
Recognizing perfect-square trinomials is the key step in "completing the square", which derives the quadratic formula itself.
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