Problem 28 - Elementary Algebra

Question

For exercises \(5-48\), simplify. $$ \frac{x^{2}}{x-7}-\frac{3 x+28}{x-7} $$

Step-by-Step Solution

Answer

The simplified expression is \( x + 4 \).

Step 1: Identify the common denominator

Both terms in the expression \( \frac{x^{2}}{x-7}-\frac{3x+28}{x-7} \) have the same denominator, which is \( x-7 \).

Step 2: Combine the numerators

Since the denominators are the same, write a single fraction with the common denominator \( x-7 \). The new expression is \( \frac{x^{2} - (3x+28)}{x-7} \)

Step 3: Distribute the subtraction

Subtract the second numerator from the first numerator: \( x^{2} - (3x + 28) = x^{2} - 3x - 28 \)

Step 4: Simplify the fraction

The simplified expression is: \( \frac{x^{2} - 3x - 28}{x-7} \)

Step 5: Factor the numerator

Factor the quadratic expression in the numerator: \( x^{2} - 3x - 28 = (x - 7)(x + 4) \)

Step 6: Cancel common factors

Cancel the \( x-7 \) term from both the numerator and the denominator: \( \frac{(x-7)(x+4)}{x-7} = x + 4 \)

Key Concepts

Common Denominator

A common denominator is crucial when working with fractions, especially in algebra. It allows you to combine multiple fractions into one by giving them a shared base. In the exercise, both fractions already had the same denominator: \(x-7\). This simplifies the process because we don't have to find a new common denominator. Instead, we can directly work with the numerators. When fractions have the same denominator, you simply borrow that common base and focus on combining the numerators.

Combining Fractions

After ensuring both fractions have the same denominator, you can combine them into a single fraction. This is done by writing the numerators together over the common denominator. For instance, in the exercise, we combined \(\frac{x^2}{x-7}\) and \(\frac{3x+28}{x-7}\) by subtracting the numerator of the second fraction from the first: \(\frac{x^2-(3x+28)}{x-7}\). Simplify the result by performing the subtraction, which in our case gave us \(\frac{x^2 - 3x - 28}{x-7}\). This step essentially merges the two fractions into one while keeping the denominator the same.

Factoring Quadratics

Factoring quadratics is a key step in simplifying many algebraic expressions. It involves breaking down a quadratic expression into a product of simpler binomials. In the exercise, the quadratic numerator \(x^2 - 3x - 28\) was factored into \((x-7)(x+4)\). How did we achieve that? We found two numbers that multiply to the constant term (-28) and add up to the coefficient of the middle term (-3). These numbers are -7 and 4. Thus, the quadratic splits into\((x-7)(x+4)\). This step makes it easier to see common factors between numerator and denominator.

Canceling Common Factors

The final step in simplifying the expression is to cancel out any common factors in the numerator and denominator. In our example, the expression \(\frac{(x-7)(x+4)}{x-7}\) allows the \(x-7\) term to be canceled, leaving us with \(x+4\). Canceling is valid here because \(x-7\) appears both in the numerator and the denominator. Always ensure that the terms are fully factored so that you don't miss out on any common terms that can be canceled. This step reduces the expression to its simplest form, making it much easier to understand and use.