##### Related Pages

# Perfect Square Trinomials

(KS4, Year 10)

## 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**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.## 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.

The brackets expand to

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 |

**a**.^{2}+ ab + ba + b^{2}# 3

Simplify the expression.

a

It does not matter which order the letters are written: ^{2}+ ab +**ba**+ b^{2}= a^{2}+ ab +**ab**+ b^{2}**ba = ab**.# 4

Add like terms.

**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.## 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.

The brackets expand to

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 |

**a**.^{2}− ab − ba + b^{2}# 3

Simplify the expression.

a

It does not matter which order the letters are written: ^{2}− ab −**ba**+ b^{2}= a^{2}− ab −**ab**+ b^{2}**ba = ab**.# 4

Subtract like terms.

**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:

**Help Us To Improve Mathematics Monster**

- Do you disagree with something on this page?
- Did you spot a typo?