Final answer:
Factoring an equation completely involves breaking it down into irreducible factors to simplify it, often to solve for unknown variables. This process may require various algebraic manipulations to facilitate factoring, always obeying basic rules of algebra to maintain equality.
Step-by-step explanation:
To factor an equation completely means to break it down into products of its factors until all of the factors are prime numbers or irreducible polynomials, depending on whether you're working with numbers or algebraic expressions. This process is essential because it allows us to find the roots or solve the equation, and it often simplifies the process of manipulating the equation further in algebraic procedures. When an equation is completely factored, no further factoring is possible.
In the context of algebra, you may see an expression or equation like ax^2 + bx + c, and to factor it completely you would look for two binomials, such as (px + q)(rx + s), that multiply to give the quadratic expression. The values of p, q, r, and s are found such that the expanded form of the binomials equals the original quadratic expression. Another common method involves recognizing special patterns like difference of squares, perfect square trinomials, or sum/difference of cubes.
When adjusting equations to factor, it may be necessary to employ techniques like simplifying coefficients, balancing the equation with whole-number coefficients, or multiplying both sides by a common factor to avoid fractions. These manipulations adhere to the fundamental rules of algebra, which maintain equality when the same operations are performed on both sides of an equation. The goal is to achieve a form where the factoring process can be applied effectively.
Consider an equation x^2 - 5x + 6 = 0. To solve for x, you would factor the quadratic into (x - 2)(x - 3) = 0. Each factor represents a potential solution to the equation, so you would set each equal to zero and solve for x, yielding x = 2 or x = 3. This is what it means to factor an equation completely and then use the factored form to solve for the unknown variable.