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 a2 + 2ab + c.

A Real Example of a Perfect Square Trinomial

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

Consider a binomial where the terms are added together.

Question

Show the perfect square trinomial shown below.

Step-by-Step:

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

Answer:

We have shown the perfect square trinomial:

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

Slider

The slider below shows another real example of perfect square trinomials.

Open the slider in a new tab

Another Real Example of a Perfect Square Trinomial

(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.

Step-by-Step:

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

Answer:

We have shown the perfect square trinomial:

(a − b)2 = a2 − 2ab + b2

Slider

The slider below shows yet another real example of perfect square trinomials.

Open the slider in a new tab

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:

See Also

What is a square number? What is the FOIL method? What is a like term? Adding like terms Subtracting like terms What is a quadratic equation?