**How to find the last term of a perfect square trinomial :**

To make any quadratic expression a perfect square, a method called completing the square may be used.

To complete the square for a quadratic expression of the form **x ^{2} + bx + missing value (or) **

Step 1 : Try to split the middle term that is coefficient of x as the multiple of 2. If it is not an even number, we should multiply and divide it by 2 to represent it as 2ab.(Here "a" and "b" are first and last terms respectively.)

Step 2 : The middle term represents the product of 2, first term and the last term.

Step 3 : The square of the last term is the missing value.

Let us look into some example problems to understand the above concept.

**Example 1 :**

Find the value of c that makes each trinomial a perfect square.

a^{2} - 12a + c

**Solution :**

= a^{2} - 12a + c

= a^{2} - 2 **⋅** a **⋅** 6 + c

Instead of "c" if we write 6^{2}, the given polynomial will become a perfect square.

= a^{2} - 2 **⋅** a **⋅** 6 + 6^{2}

Since the above polynomial is in the form a^{2} - 2ab + b^{2}, we can write it as (a + b)^{2}

= (a + 6)^{2}

Hence the value of c is 6^{2}, that is 36.

**Example 2 :**

Find the value of c that makes each trinomial a perfect square.

x^{2} - 16x + c

**Solution :**

= x^{2} - 16x + c

= x^{2} - 2 **⋅** x **⋅** 8 + c

Instead of "c" if we write 8^{2}, the given polynomial will become a perfect square.

= x^{2} - 2 **⋅** x **⋅** 8 + 8^{2}

Since the above polynomial is in the form a^{2} - 2ab + b^{2}, we can write it as (a - b)^{2}

= (x - 8)^{2}

Hence the value of c is 8^{2}, that is 64.

**Example 3 :**

Find the value of c that makes each trinomial a perfect square.

x^{2} - 10x + c

**Solution :**

= x^{2} - 10x + c

= x^{2} - 2 **⋅** x **⋅** 5 + c

Instead of "c" if we write 5^{2}, the given polynomial will become a perfect square.

= x^{2} - 2 **⋅** x **⋅** 5 + 5^{2}

Since the above polynomial is in the form a^{2} - 2ab + b^{2}, we can write it as (a - b)^{2}

= (x - 5)^{2}

Hence the value of c is 5^{2}, that is 25.

**Example 4 :**

Find the value of c that makes each trinomial a perfect square.

x^{2} - 7x + c

**Solution :**

= x^{2} - 7x + c

= x^{2} - (2/2) **⋅** 7 **⋅ **x + c

= x^{2} - 2⋅ x ⋅ (7/2) + c

Instead of "c" if we write (7/2)^{2}, the given polynomial will become a perfect square.

= x^{2} - 2⋅ x ⋅ (7/2) + (7/2)^{2}

Since the above polynomial is in the form a^{2} - 2ab + b^{2}, we can write it as (a - b)^{2}

= (x - (7/2))^{2}

Hence the value of c is (7/2)^{2}, that is 49/4.

**Example 5 :**

Find the value of c that makes each trinomial a perfect square.

x^{2} + 11x + c

**Solution :**

= x^{2} + 11x + c

= x^{2} - (2/2) **⋅** 11 **⋅ **x + c

= x^{2} - 2⋅ x ⋅ (11/2) + c

Instead of "c" if we write (11/2)^{2}, the given polynomial will become a perfect square.

= x^{2} - 2⋅ x ⋅ (11/2) + (11/2)^{2}

Since the above polynomial is in the form a^{2} - 2ab + b^{2}, we can write it as (a - b)^{2}

= (x - (11/2))^{2}

Hence the value of c is (11/2)^{2}, that is 121/4.

We hope that the students would have understood the stuff and example problems explained on "How to find the last term of a perfect square trinomial".

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