Question

How do you factor ${{x}^{2}}-{{y}^{2}}+{{z}^{2}}-2xz$ ?

Answer 1

Hint: In this question, we have to find the factors of an algebraic equation. So, we will apply the algebraic identities and split the middle terms method to get the solution. We first rewrite the equation by taking x and z variables together, then we will apply splitting the middle term method, that is we split the middle term as the sum of $-z$ and $-z$ , and then take common x from the first two terms and z from the last two terms. On necessary calculations, we will apply the algebraic identity $(a-b)(a+b)={{a}^{2}}-{{b}^{2}}$ in the equation, to get the required result to the problem.

Complete step-by-step solution:
According to the question, we have to find the factors of an algebraic equation.
So, we will apply splitting the middle term method and the algebraic identity to get the result.
The equation given to us is ${{x}^{2}}-{{y}^{2}}+{{z}^{2}}-2xz$ ---------------- (1)
Now, we first rewrite equation (1) as taking x and z variables together, that is
$({{x}^{2}}+{{z}^{2}}-2xz)-{{y}^{2}}$
On further simplification, we get
$({{x}^{2}}-2xz+{{z}^{2}})-{{y}^{2}}$
Now, we get a quadratic equation in the above equation, so we will apply the splitting the middle term method in the above equation, that is we split the middle term of the quadratic equation as the sum of –zx and –zx, we get
$({{x}^{2}}-xz-xz+{{z}^{2}})-{{y}^{2}}$
Now, take common x from the first two terms and –z from the last two terms, we get
$(x(x-z)-z(x-z))-{{y}^{2}}$
So, we will take common (x-z) from the above equation, we get
$((x-z)(x-z))-{{y}^{2}}$
Therefore, we get
${{(x-z)}^{2}}-{{y}^{2}}$
Now, we will apply the algebraic identity $(a-b)(a+b)={{a}^{2}}-{{b}^{2}}$ in the above equation, we get
$(x-z-y)(x-z+y)$
Therefore, for the algebraic equation ${{x}^{2}}-{{y}^{2}}+{{z}^{2}}-2xz$ , its factors are $(x-z-y)(x-z+y)$ .

Note: While solving this equation, do mention all the identities and formulas properly to avoid any confusion and mathematical errors. One of the alternative methods to solve this problem is to take the factors of the given equation without rewriting them, and then apply the splitting the middle term method, to get the required result to the problem.