Question

Simplify:
 $ {{x}^{2}}+{{z}^{2}}-2xz $

Answer 1

Hint: Recall the identity: $ {{(a\pm b)}^{2}}={{a}^{2}}\pm 2ab+{{b}^{2}} $ .
In order to factorize a quadratic expression, Split the term consisting of the product of the variables, $ -2xz $ in this case, into a sum of two terms whose product is equal to the product of the remaining two terms $ {{x}^{2}}{{z}^{2}} $.
Separate the common factors from both the pairs of terms.

Complete step-by-step answer:
The given expression $ {{x}^{2}}+{{z}^{2}}-2xz $ is a quadratic expression. Let us split its term $ -2xz $ into $ -xz $ and $ -xz $ , such that their product is equal to $ {{x}^{2}}{{z}^{2}} $ , the product of the other two terms.
∴ $ {{x}^{2}}+{{z}^{2}}-2xz $
= $ {{x}^{2}}-xz-xz+{{z}^{2}} $
Separating the common factors from the first two and the last two terms, we get:
= $ x(x-z)-z(x-z) $
Separating the common factor $ (x-z) $ from both the terms, we get:
= $ (x-z)(x-z) $
Which can be written as:
= $ {{(x-z)}^{2}} $ , which is the required simplification.

Note: It is not always possible to simplify a given expression.
e.g. $ {{a}^{2}}+{{b}^{2}}+3ab $
Some useful algebraic identities:
 $ {{(a-b)}^{2}}={{(b-a)}^{2}} $
 $ (a+b)(a-b)={{a}^{2}}-{{b}^{2}} $
 $ {{(a\pm b)}^{2}}={{a}^{2}}\pm 2ab+{{b}^{2}} $
 $ {{(a\pm b)}^{3}}={{a}^{3}}\pm 3ab(a\pm b)\pm {{b}^{3}} $
 $ (a\pm b)({{a}^{2}}\mp ab+{{b}^{2}})={{a}^{3}}\pm {{b}^{3}} $