Question

Simplify the expression, if possible. $$ \frac{x^2-3 x-18}{x^2-7 x+6} $$

Short answer

The simplified expression is \(\frac{x+3}{x-1}\).

Step-by-step solution

Factorize the Quadratic Expressions

The quadratic expressions \(x^2-3x-18\) and \(x^2-7x+6\) in the numerator and the denominator respectively can be factorized by finding two numbers that multiply to give the product of the coefficients of the \(x^2\) and the constant term, and sum up to the coefficient of \(x\). For the numerator \(x^2-3x-18\), the numbers are 6 and -3. This therefore factorizes to \((x-6)(x+3)\). For the denominator \(x^2-7x+6\), the numbers are -1 and -6, hence it factorizes to \((x-1)(x-6)\). This leaves us with the expression \(\frac{(x-6)(x+3)}{(x-1)(x-6)}\).

Simplify the Rational Expression

To simplify the expression, cancel out the common factors in the numerator and denominator. In this case, the (x-6) term can be cancelled out. This leaves us with the simplified expression \(\frac{x+3}{x-1}\).

Key concepts

Factoring Quadratic Expressions

Factoring quadratic expressions is an essential skill in algebra. It enables you to break down complex polynomials into simpler terms, making them easier to manage and solve. When we factorize a quadratic expression such as \( x^2 - 3x - 18 \), our goal is to express it in the form of \((x - p)(x - q)\), where \(p\) and \(q\) are numbers that multiply to the constant term and sum to the coefficient of \(x\).

• Look for two numbers that multiply to the product of the quadratic coefficient (1 here) and the constant term (-18), and also add to the linear coefficient (-3).
• For \(x^2 - 3x - 18\), these numbers are 6 and -3, resulting in the factorization \((x - 6)(x + 3)\).
• Similarly, for \(x^2 - 7x + 6\), finding the numbers -1 and -6 gives us \((x - 1)(x - 6)\).

Mastering this process is crucial as it lays the groundwork for simplifying rational expressions properly.

Canceling Common Factors

Once we have a rational expression in a factorized form, such as \( \frac{(x-6)(x+3)}{(x-1)(x-6)} \), the next step is to simplify by canceling out common factors. This is a fundamental concept because it reduces a fraction to its simplest form.

• Identify the terms that appear in both the numerator and the denominator. In this example, \((x-6)\) appears in both.
• Cancel out these common terms, provided they don't equate to zero in the domain of the variable \(x\).
• This process results in the simpler expression: \( \frac{x+3}{x-1} \).

Be cautious of the restrictions on the variable—canceling terms such as \((x-6)\) implies \(x eq 6\). This ensures the original expression’s denominator doesn't become zero, maintaining its validity.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing a complex fraction to its simplest form, making it easier to work with or solve. The steps include factoring and canceling, but it must be approached methodically to avoid mistakes.

• The expression \( \frac{x^2-3x-18}{x^2-7x+6} \) first needs to be broken down into simpler parts through factoring.
• Once factorized, each term's structure becomes clear, revealing common factors that can be canceled.
• After canceling the common factors, check for any restrictions (such as \(x eq 6\) and \(x eq 1\)), derived from the original denominator to avoid undefined expressions.

This method ensures the fraction remains equivalent to the original while being presented in its most straightforward form, aiding in further algebraic operations or applications.