**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

**a**² + 2 ab + b² = (a + b)²

By comparing the above expression ** **x² + 2 (6) x + 6² with the formula ** a**² + 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

**a**² + 2 ab + b² = (a + b)²

By comparing the above expression ** **a² + 2 ⋅ a ⋅ 2 + 2² with the formula ** a**² + 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 **a**² + 2 ab + b²

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

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