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.

9th Grade Math

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