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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