## The Lesson

A perfect square trinomial is the result of squaring a binomial.

• A binomial is two terms added (or subtracted) together. In the example above, the binomial is a + b.
• A squared binomial means multiplying the binomial by itself. In our example, the squared binomial is (a + b)2.
(a + b)2 = (a + b) × (a + b)
• A trinomial is three terms added (or subtracted) together. In our example, the trinomial is a2 + 2ab + c.

## (a + b)2 = a2 + 2ab + b2

Consider a binomial where the terms are added together.

## Question

Show the perfect square trinomial shown below.

# 1

Square the brackets by writing the squared binomial as two brackets multiplied together.
(a + b)2 = (a + b) × (a + b) = (a + b)(a + b)
Don't forget: 2 means squared and writing letters or brackets next to each other means they are multiplying each other.

# 2

Use the FOIL method to expand the brackets.
 a2 Firsts $$\:\:\:\:\:\:\:\:\:\:\:\:$$ (a + b)(a + b) $$\:\:\:\:\:\:\:\:\:\:\:\:$$ a × a a2 + ab Outsides $$\:\:\:\:\:\:$$ (a + b)(a + b) $$\:\:\:\:\:\:\:\:\:\:\:\:$$ a × b a2 + ab + ba Insides $$\:\:\:\:\:\:\:\:\:$$ (a + b)(a + b) $$\:\:\:\:\:\:\:\:\:\:\:\:$$ b × a a2 + ab + ba + b2 Lasts $$\:\:\:\:\:\:\:\:\:\:\:\:$$ (a + b)(a + b) $$\:\:\:\:\:\:\:\:\:\:\:\:$$ b × b
The brackets expand to a2 + ab + ba + b2.

# 3

Simplify the expression.
a2 + ab + ba + b2 = a2 + ab + ab + b2
It does not matter which order the letters are written: ba = ab.

# 4

Add like terms. ab and ab are like terms. Add them together:
a2 + ab + ab + b2 = a2 + 2ab + b2

We have shown the perfect square trinomial:
(a + b)2 = a2 + 2ab + b2

## Lesson Slides

The slider below shows another real example of perfect square trinomials. Open the slider in a new tab

## (a − b)2 = a2 − 2ab + b2

Consider a binomial where the terms are subtracted from each other.

## Question

Show the perfect square trinomial shown below.

# 1

Square the brackets by writing the squared binomial as two brackets multiplied together.
(a − b)2 = (a − b) × (a − b) = (a − b)(a − b)
Don't forget: 2 means squared and writing letters or brackets next to each other means they are multiplying each other.

# 2

Use the FOIL method to expand the brackets.
 a2 Firsts $$\:\:\:\:\:\:\:\:\:\:\:\:$$ (a − b)(a − b) $$\:\:\:\:\:\:\:\:\:\:\:\:$$ a × a a2 − ab Outsides $$\:\:\:\:\:\:$$ (a − b)(a − b) $$\:\:\:\:\:\:\:\:\:\:\:\:$$ a × −b a2 − ab − ba Insides $$\:\:\:\:\:\:\:\:\:$$ (a − b)(a − b) $$\:\:\:\:\:\:\:\:\:\:\:\:$$ −b × a a2 − ab − ba + b2 Lasts $$\:\:\:\:\:\:\:\:\:\:\:\:$$ (a − b)(a − b) $$\:\:\:\:\:\:\:\:\:\:\:\:$$ −b × −b
The brackets expand to a2 − ab − ba + b2.

# 3

Simplify the expression.
a2 − ab − ba + b2 = a2 − ab − ab + b2
It does not matter which order the letters are written: ba = ab.

# 4

Subtract like terms. ab and ab are like terms. Subtract them from each other:
a2 − ab − ab + b2 = a2 − 2ab + b2