Perfect Square Trinomials –
A trinomial which is the square of a binomial is called a perfect square trinomial. Two examples of such trinomials are x²+ 2xy + y² [=(x+y)²] and x²- 2xy + y² [=(x-y)²]
In a Perfect Square Trinomial –
a) Two terms are perfect squares
b) The third term (which may be positive or negative) is twice the product of the square roots of the other two terms.
There are some examples are given below for your better understanding –
Example.1) 9a² + 12a + 4
here, 9a² = (3a)², 4 = 2² and 12a = 2 X 3a X 2 = twice the product of the square roots of the other two terms.
Thus, this is a perfect square trinomial that can be expressed as the square of a binomial as follows -
9a² + 12a + 4
= (3a)²+ 2 X 3a X 2 + (2)² = (3a + 2)² (Ans.)
Example.2) 81a² - 72a + 16
here, 81a² = (9a)², 16 = 4² and 72a = 2 X 9a X 4 = twice the product of the square roots of the other two terms.
Thus, this is a perfect square trinomial that can be expressed as the square of a binomial as follows -
81a² - 72a + 16
= (9a)²- 2 X 9a X 4 + (4)² = (9a - 4)² (Ans.)
Example.3) 36a² - 36ab + 9b²
here, 36a² = (6a)², 9b = (3b)² and 36ab = 2 X 6a X 3b = twice the product of the square roots of the other two terms.
Thus, this is a perfect square trinomial that can be expressed as the square of a binomial as follows -
36a² - 36ab + 9b²
= (6a)²- 2 X 6a X 3b + (3b)² = (6a – 3b)² (Ans.)
a² 2ab b²
Example.4) -------- + --------- + ---------
9 21 49
a² a b² b
here, ------ = (------)², ------- = (-------)² and
9 3 49 7
------- = 2 X ------- X ------- = twice the product of the square
21 3 7
roots of the other two terms.
Thus, this is a perfect square trinomial that can be expressed as the square of a binomial as follows -
a² 2ab b²
---------- + ---------- + -----------
9 21 49
a a b b
= (-------)² + 2 X ------- X ------- + (-------)²
3 3 7 7
a b
= (------- + -------)² (Ans.)
3 7