Perfect Square Trinomials
(KS4, Year 10)
A perfect square trinomial is the result of squaring a binomial.

perfect square trinomial
  • 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.
perfect square trinomial a plus b

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

Lesson Slides

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

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.
perfect square trinomial a minus b

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

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:

perfect square trinomial quadratic equation add If our binomial is not a − b but x − a, then the expansion is also a quadratic equation:

perfect square trinomial quadratic equation minus
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This page was written by Stephen Clarke.