What Is a Perfect Square Trinomial?
What Is a Perfect Square Trinomial?
A perfect square trinomial is the result of squaring a binomial.
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A binomial is two terms added (or subtracted) together.
In the example above, the binomial is a + b.
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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)
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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
Answer:
We have shown the perfect square trinomial:
(a + b)2 = a2 + 2ab + b2
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
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:
If our binomial is not a − b but x − a, then the expansion is also a quadratic equation:
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