By writing the given quadratic trinomial in the form of a^{2} + 2ab + b^{2} or a^{2} - 2ab + b^{2}, we can find the constant term by comparing the unknown with b^{2}.
Step 1 :
If the leading coefficient is other than 1, we have to factor that out from all the terms.
Step 2 :
Write the coefficient of x as multiple of 2.
Step 3 :
Now that will be in any one of the forms,
a^{2} + 2ab + b^{2} or a^{2} - 2ab + b^{2}
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.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM