What Is a Perfect Square Trinomial?
What Is a Perfect Square Trinomial?
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 a^{2} + 2ab + c.
A Real Example of a Perfect Square Trinomial
(a + b)^{2} = a^{2} + 2ab + b^{2}
Consider a binomial where the terms are added together.
Question
Show the perfect square trinomial shown below.
StepbyStep:
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.
a^{2}  Firsts \(\:\:\:\:\:\:\:\:\:\:\:\:\) (a + b)(a + b) \(\:\:\:\:\:\:\:\:\:\:\:\:\) a × a 
a^{2} + ab  Outsides \(\:\:\:\:\:\:\) (a + b)(a + b) \(\:\:\:\:\:\:\:\:\:\:\:\:\) a × b 
a^{2} + ab + ba  Insides \(\:\:\:\:\:\:\:\:\:\) (a + b)(a + b) \(\:\:\:\:\:\:\:\:\:\:\:\:\) b × a 
a^{2} + ab + ba + b^{2}  Lasts \(\:\:\:\:\:\:\:\:\:\:\:\:\) (a + b)(a + b) \(\:\:\:\:\:\:\:\:\:\:\:\:\) b × b 
The brackets expand to a^{2} + ab + ba + b^{2}.
3
Simplify the expression.
a^{2} + ab + ba + b^{2} = a^{2} + ab + ab + b^{2}
It does not matter which order the letters are written: ba = ab.
4
ab and ab are like terms. Add them together:
a^{2} + ab + ab + b^{2} = a^{2} + 2ab + b^{2}
Answer:
We have shown the perfect square trinomial:
(a + b)^{2} = a^{2} + 2ab + b^{2}
Another Real Example of a Perfect Square Trinomial
(a − b)^{2} = a^{2} − 2ab + b^{2}
Consider a binomial where the terms are subtracted from each other.
Question
Show the perfect square trinomial shown below.
StepbyStep:
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.
a^{2}  Firsts \(\:\:\:\:\:\:\:\:\:\:\:\:\) (a − b)(a − b) \(\:\:\:\:\:\:\:\:\:\:\:\:\) a × a 
a^{2} − ab  Outsides \(\:\:\:\:\:\:\) (a − b)(a − b) \(\:\:\:\:\:\:\:\:\:\:\:\:\) a × −b 
a^{2} − ab − ba  Insides \(\:\:\:\:\:\:\:\:\:\) (a − b)(a − b) \(\:\:\:\:\:\:\:\:\:\:\:\:\) −b × a 
a^{2} − ab − ba + b^{2}  Lasts \(\:\:\:\:\:\:\:\:\:\:\:\:\) (a − b)(a − b) \(\:\:\:\:\:\:\:\:\:\:\:\:\) −b × −b 
The brackets expand to a^{2} − ab − ba + b^{2}.
3
Simplify the expression.
a^{2} − ab − ba + b^{2} = a^{2} − ab − ab + b^{2}
It does not matter which order the letters are written: ba = ab.
4
ab and ab are like terms. Subtract them from each other:
a^{2} − ab − ab + b^{2} = a^{2} − 2ab + b^{2}
Answer:
We have shown the perfect square trinomial:
(a − b)^{2} = a^{2} − 2ab + b^{2}
Perfect Square Trinomials and Quadratic Equations
A perfect square trinomial expands to a quadratic equation.
If our binomial is not a + b but x + a, then the expansion is a quadratic equation:
If our binomial is not a − b but x − a, then the expansion is also a quadratic equation: