HOW TO FIND THE LAST TERM OF A PERFECT SQUARE TRINOMIAL

To make any trinomial as a perfect square, we use the method called completing the square.

To complete the square of a quadratic expression in the form

a2 + 2ab + missing value

(or) 

a2 - 2ab + missing value,

we can follow the steps below. 

Step 1 :

Write the middle term as a 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 has to represent the product of 2, first term and the last term.

Step 3 :

The square of the last term is the missing value.

In both

a2 + 2ab + missing value

and 

a2 - 2ab + missing value,

the missing value is b2.

Because b2 at the missing value makes the above trinomials as perfect squares.  

That is, 

a2 + 2ab + b2  =  (a + b)2

a2 - 2ab + b2  =  (a - b)2

Example 1 :

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

a2 - 12a + c

Solution :

=  a2 - 12a + c

=  a2 - 2 ⋅ a ⋅ 6 + c

Instead of "c" if we write 62, the given polynomial will become a perfect square.

Then,

=  a2 - 2 ⋅ a ⋅ 6 + 62

Since the above polynomial is in the form a2 - 2ab + b2, we can write it as (a + b)2

=  (a + 6)2

So, the value of c is 62, that is 36.

Example 2 :

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

x2 - 16x + c

Solution :

=  x2 - 16x + c

=  x2 - 2 ⋅ x ⋅ 8 + c

Instead of "c" if we write 82, the given polynomial will become a perfect square.

=  x2 - 2 ⋅ x ⋅ 8 + 82

Since the above polynomial is in the form a2 - 2ab + b2, we can write it as (a - b)2

=  (x - 8)2

So, the value of c is 82, that is 64.

Example 3 :

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

x2 - 10x + c

Solution :

=  x2 - 10x + c

=  x2 - 2 ⋅ x ⋅ 5 + c

Instead of "c" if we write 52, the given polynomial will become a perfect square.

=  x2 - 2 ⋅ x ⋅ 5 + 52

Since the above polynomial is in the form a2 - 2ab + b2, we can write it as (a - b)2

=  (x - 5)2

So, the value of c is 52, that is 25.

Example 4 :

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

x2 - 7x + c

Solution :

=  x2 - 7x + c

=  x2 - (2/2) ⋅ 7 ⋅ x + c

=  x2 - 2⋅ x ⋅ (7/2) + c

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

=  x2 - 2⋅ x ⋅ (7/2) + (7/2)2

Since the above polynomial is in the form a2 - 2ab + b2, we can write it as (a - b)2

=  (x - (7/2))2

So, 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. 

x2 + 11x + c

Solution :

=  x2 + 11x + c

=  x2 - (2/2) ⋅ 11 ⋅ x + c

=  x2 - 2⋅ x ⋅ (11/2) + c

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

=  x2 - 2⋅ x ⋅ (11/2) + (11/2)2

Since the above polynomial is in the form a2 - 2ab + b2, we can write it as (a - b)2

=  (x - (11/2))2

So, the value of c is (11/2)2, that is 121/4.

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