FIND THE CONSTANT TERM OF TRINOMIAL USING COMPLETING THE SQUARE

Find the value of c that makes the expression a perfect square trinomial. Then write the expression as a square of a binomial.

Problem 1 :

x² + 8x + c

Solution :

= x² + 8x + c

= x² + 2 ∙ x ∙ 4 + c

Instead of ‘c’ if we write 4², the given polynomial will become a perfect square.

 = x² + 2 ∙ x ∙ 4 + 4²

Since the above polynomial is in form a² + 2ab + b², we can write it as

(a + b)² = (x + 4)²

So, the value of c is 4², that is 16.

Problem 2 :

x² - 22x + c

Solution :

= x² - 22x + c

= x² - 2 ∙ x ∙ 11 + c

Instead of ‘c’ if we write 11², the given polynomial will become a perfect square.

 = x² - 2 ∙ x ∙ 11 + 11²

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

(a - b)² = (x - 11)²

So, the value of c is 11², that is 121.

Problem 3 :

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 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.

Problem 4 :

x² + 3x + c

Solution :

= x² + 3x + c

= x² + (2/2) ∙ x ∙ 3 + c

= x² + 2 ∙ x ∙ (3/2) + c

Instead of ‘c’ if we write (3/2)², the given polynomial will become a perfect square.

 = x² + 2 ∙ x ∙ (3/2) + (3/2)²

Since the above polynomial is in form a² + 2ab + b², we can write it as

(a + b)² = (x + (3/2))²

So, the value of c is (3/2)², that is 9/4.

Problem 5 :

x² - 9x + c

Solution :

= x² - 9x + c

= x² - (2/2) ∙ x ∙ 9 + c

= x² - 2 ∙ x ∙ (9/2) + c

Instead of ‘c’ if we write (9/2)², the given polynomial will become a perfect square.

 = x² - 2 ∙ x ∙ (9/2) + (9/2)²

Since the above polynomial is in form a² - 2ab + b², we can write it as (a - b)² = (x - (9/2))²

So, the value of c is (9/2)², that is 81/4.

Problem 6 :

9x² -12x + c

Solution :

9x² - 12x + c

= x² - (12/9)x + c

= x² - (4/3)x + c

= x² - 2 ∙ (2/3)x + c

Instead of ‘c’ if we write (2/3)², the given polynomial will become a perfect square.

= x² - 2 ∙ (2/3) ∙ x + (2/3)²

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

So, the value of c is (2/3)², that is 4/9.