91Ó°ÊÓ

In algebra, factoring is the process of breaking down an expression into products of simpler expressions. This is a critical step for simplifying complex equations and making them easier to solve.
In the example given, we see the equation: \[xy^2 + xz^2 = xw^2 - 6\]Notice how each term on the left-hand side has a common factor of 'x'. By factoring out 'x', you transform the equation:
\[x(y^2 + z^2) = xw^2 - 6\]Now, it's easier to identify and manipulate the remaining parts of the expression. Factorization often simplifies equations and reveals relationships between terms that might not be immediately apparent.

Isolating the variable means rearranging an equation so that the variable you want to solve for is by itself on one side of the equation. This involves moving terms around using basic algebraic operations like addition, subtraction, multiplication, and division. In our example: \[x(y^2 + z^2) = xw^2 - 6\]We begin by isolating 'x' by first moving terms involving 'x' to one side:
\[x(y^2 + z^2) - xw^2 = -6\]Now, we group 'x' together (observe how 'x' is a common factor on the left side):
\[x(y^2 + z^2 - w^2) = -6\]Next, to fully isolate 'x', we divide both sides by the remaining expression tied to 'x':
\[x = \frac{-6}{y^2 + z^2 - w^2}\]So, 'x' is by itself, and we have isolated it effectively.

Algebraic manipulation involves using arithmetic operations to rearrange and simplify expressions and equations. This can include factoring, distributing, combining like terms, and isolating variables. It's all about transforming the equation step-by-step to get the desired form. Let’s revisit our equation: \[xy^2 + xz^2 = xw^2 - 6\]Here, we started by factoring out 'x':
\[x(y^2 + z^2) = xw^2 - 6\]Then, we reorganized the terms to isolate 'x':
\[x(y^2 + z^2) - xw^2 = -6\]Next, we factored 'x' out again from the left-hand side:
\[x (y^2 + z^2 - w^2) = -6\]Finally, we divided both sides by the remaining expression to solve for 'x':
\[x = \frac{-6}{y^2 + z^2 - w^2}\]Each step is a form of algebraic manipulation to simplify and solve for the variable. Mastering these techniques makes solving equations more intuitive and manageable.