# Factorization of a Perfect-square Trinomial

Here we will learn the process of Factorization of a Perfect-square Trinomial.

A trinomial of the form a2 ± 2ab + b2 = (a ± b)2 = (a ± b)(a ± b)

Solved examples on Factorization of a Perfect-square Trinomial

1. Factorize: x2 + 6x + 9

Solution:

Here, given expression = x$$^{2}$$ + 6x + 9

= x$$^{2}$$ + 2 ∙ x ∙ 3 + 3$$^{2}$$

= (x + 3)$$^{2}$$

= (x + 3)(x + 3)

2. Factorize: x$$^{2}$$ + x + ¼

Solution:

Here, given expression = x$$^{2}$$ + x + ¼

= x$$^{2}$$ + 2 ∙ x ∙ $$\frac{1}{2}$$ + ($$\frac{1}{2}$$)$$^{2}$$

= (x + $$\frac{1}{2}$$)$$^{2}$$

= (x + $$\frac{1}{2}$$)(x + $$\frac{1}{2}$$)

3. Factorize: 25m$$^{2}$$ – 10m + 1

Solution:

Here, given expression = 25m$$^{2}$$ – 10m + 1

= (5m)$$^{2}$$ – 2 ∙ 5m ∙ 1 + 1$$^{2}$$

= (5m – 1)$$^{2}$$

= (5m – 1)(5m – 1)

4. Factorize: 4a$$^{2}$$ – 4ab + b$$^{2}$$

Solution:

Here, given expression = 4a$$^{2}$$ – 4ab + b$$^{2}$$

= (2a)$$^{2}$$ – 2 ∙ 2a ∙ b + b$$^{2}$$

= (2a – b)$$^{2}$$

= (2a – b)(2a – b)

5. Factorize: z$$^{2}$$ + $$\frac{1}{z^{2}}$$ – 2.

Solution:

Here, given expression = z$$^{2}$$ + $$\frac{1}{z^{2}}$$ – 2

= z$$^{2}$$ - 2 ∙ z ∙ $$\frac{1}{z}$$ + ($$\frac{1}{z}$$)$$^{2}$$

= (z - $$\frac{1}{z^{2}}$$)$$^{2}$$

= (z - $$\frac{1}{z^{2}}$$)(z - $$\frac{1}{z^{2}}$$).

6. Factorize: 25m$$^{2}$$ + $$\frac{5m}{2}$$ + $$\frac{1}{16}$$.

Solution:

Here, given expression = 25m$$^{2}$$ + $$\frac{5m}{2}$$ + $$\frac{1}{16}$$.

= (5m)$$^{2}$$ + $$\frac{5m}{2}$$ + ($$\frac{1}{4}$$)$$^{2}$$, [Two terms should be such that they are squares]

= (5m)$$^{2}$$ + 2 ∙ 5m ∙ $$\frac{1}{4}$$ + ($$\frac{1}{4}$$)$$^{2}$$ [The third term should be twice the product of the terms whose squares are the other two terms]

= (5m + $$\frac{1}{4}$$)$$^{2}$$

= (5m + $$\frac{1}{4}$$)(5m + $$\frac{1}{4}$$)

Note: The trinomial ax$$^{2}$$ + bx + c is a perfect square if b$$^{2}$$ = 4ac.