# HOW TO DETERMINE WHETHER THE GIVEN TRINOMIAL IS A PERFECT SQUARE

## About "How to determine whether the given trinomial is a perfect square"

How to determine whether the given trinomial is a perfect square :

If the given trinomial is a perfect square then it has to be written as product of two same factors.

To test whether the given trinomial is a perfect square, we should try to write the trinomial in the form of

a² + 2ab + b² (or) a² - 2ab + b²

If we are able to write the given trinomial in the above form, then it is perfect square.

Otherwise the given trinomial is not a perfect square.

Example 1 :

Determine whether each trinomial is a perfect square trinomial. If so, factor it.

x² + 12x + 36

Solution :

=  x² + 12x + 36

=  x² + 2 (6) x + 6²

a² + 2 ab + b²  =  (a + b)²

By comparing the above expression  x² + 2 (6) x + 6² with the formula  a² + 2 ab + b², instead of "a" we have "x"and instead of "b" we have "6".

x² + 2 (6) x + 6²  =  (x + 6)²

=  (x + 6) (x + 6)

Hence, the given trinomial x² + 12x + 36 is a perfect square.

Example 2 :

Determine whether each trinomial is a perfect square trinomial. If so, factor it.

n²  - 13x + 36

Solution :

=  n²  - 13x + 36

Since the middle term is odd, we cannot split this as the multiple of 2. Hence it is not a perfect square.

Example 3 :

Determine whether each trinomial is a perfect square trinomial. If so, factor it.

a² + 4a + 4

Solution :

=   a² + 4a + 4

=   a² + 2  a  2 + 2²

a² + 2 ab + b²  =  (a + b)²

By comparing the above expression  a² + 2  a  2 + 2² with the formula  a² + 2 ab + b², instead of "a" we have "a"and instead of "b" we have "2".

a² + 2  a  2 + 2²  =  (a + 2)²

=  (a + 2) (a + 2)

Hence, the given trinomial  a² + 4a + 4 is a perfect square.

Example 4 :

Determine whether each trinomial is a perfect square trinomial. If so, factor it.

x² - 10x + 100

Solution :

=    x² - 10x + 100

=    x² - 2 x (5) + 10²

Instead of "b", we have 5 but the last term is not 5². Hence the given trinomial  x² - 10x + 100 is not a perfect square.

Example 5 :

Determine whether each trinomial is a perfect square trinomial. If so, factor it.

2n² + 17n + 21

Solution :

Since the middle term is odd, we cannot split 17 as the multiple of 2. So, we should not write the above expression in the form of a² + 2 ab + b²

Hence, the given trinomial 2n² + 17n + 21 is not a perfect square.

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