Question

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

Answer 1

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:
${{(a-b)}^{2}}={{(b-a)}^{2}}$
$(a+b)(a-b)={{a}^{2}}-{{b}^{2}}$
${{(a\pm b)}^{2}}={{a}^{2}}\pm 2ab+{{b}^{2}}$
${{(a\pm b)}^{3}}={{a}^{3}}\pm 3ab(a\pm b)\pm {{b}^{3}}$
$(a\pm b)({{a}^{2}}\mp ab+{{b}^{2}})={{a}^{3}}\pm {{b}^{3}}$