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.

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

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

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

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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.

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

ab and ab are like terms. Subtract them from each other:

a2 − ab − ab + b2 = a2 − 2ab + b2

We have shown the perfect square trinomial:

(a − b)2 = a2 − 2ab + b2

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

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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: