Simplify (3x-1)/(x^2-49)
The given problem asks to perform algebraic simplification on the expression (3x-1)/(x^2-49). Simplifying this expression typically involves factoring the denominator, identifying and canceling out common factors (if any) in the numerator and denominator, and expressing the result in its simplest form. Simplification may also involve applying algebraic rules, such as the difference of squares in the denominator.
$\frac{3 x - 1}{x^{2} - 49}$
Answer
Solution:
Step 1:
Express $49$ as the square of $7$. Thus, the expression becomes $\frac{3x - 1}{x^2 - 7^2}$.
Step 2:
Recognize that the denominator is a difference of squares. Apply the formula $a^2 - b^2 = (a + b)(a - b)$, where $a = x$ and $b = 7$. This gives us $\frac{3x - 1}{(x + 7)(x - 7)}$.
Knowledge Notes:
The problem involves simplifying a rational expression. To do this, we can factor the denominator and simplify the expression if possible. Here are the relevant knowledge points:
Difference of Squares: This is a pattern where you have two terms that are squared and subtracted from each other, $a^2 - b^2$. It can be factored into $(a + b)(a - b)$.
Factoring: This is the process of breaking down an expression into a product of simpler expressions. In the case of the difference of squares, we are looking for two binomials that multiply to give the original quadratic expression.
Simplifying Rational Expressions: Once the denominator is factored, we look for common factors in the numerator and the denominator that can be canceled out. However, in this problem, there are no common factors to cancel.
LaTeX Formatting: When presenting mathematical expressions, LaTeX is used to format the expressions for clarity. For example, exponents are written using the caret symbol ^, and fractions are written using the \frac{}{} command.
By applying the difference of squares formula, we can rewrite the denominator but cannot further simplify the expression since the numerator and the denominator do not have common factors.
