Problem 15 - Elementary Algebra

Question

For exercises \(5-48\), simplify. $$ \frac{r^{2}-12 r}{r+2}-\frac{28}{r+2} $$

Step-by-Step Solution

Answer

\( r - 14 \)

Step 1: Identify common denominators

Observe that both terms of the expression have the same denominator, which is \( r+2 \). This makes it easier to combine the fractions.

Step 2: Combine the fractions

Combine the fractions using the common denominator: \(\frac{r^2 - 12r}{r+2} - \frac{28}{r+2} = \frac{r^2 - 12r - 28}{r+2}\).

Step 3: Simplify the numerator

Factorize the quadratic expression in the numerator. Note that \(r^2 - 12r - 28\) can be factored as follows: \( (r - 14)(r + 2)\).

Step 4: Simplify the entire fraction

Substitute the factored form back into the fractions: \( \frac{(r - 14)(r + 2)}{r + 2}\). Then cancel the common factor \(r + 2\) in the numerator and denominator.

Step 5: Write the final simplified expression

After canceling out the common factors, the expression simplifies to: \( r - 14 \).

Key Concepts

Factoring Quadratics

Factoring quadratics is a method used to simplify quadratic expressions, typically in the form of \( ax^2 + bx + c \). This can be done by finding two numbers that multiply to ac (the product of a and c) and add up to b. For our original numerator, we started with \( r^2 - 12r - 28 \). To factor it:

• Identify the numbers whose product is -28 (since -28 is the constant term) and whose sum is -12 (since -12 is the linear coefficient).
• The numbers -14 and 2 come to mind, because \(-14 \times 2 = -28\) and \(-14 + 2 = -12\).

So, we rewrite the quadratic as: \( r^2 - 14r + 2r - 28 \). Next, we group and factor by grouping:

• \( r(r - 14) + 2(r - 14) \)
• \( (r - 14)(r + 2) \)

After factoring, we arrive at the product \( (r - 14)(r + 2) \).

Common Denominators

When combining fractions, having a common denominator is essential. In our exercise, both fractions share the denominator \( r + 2 \), making it straightforward to combine them.

• For fractions, the common denominator is the number below the fraction line that both fractions share.
• In expressions, always look for common denominators to simplify your work.

Here, the expression is \( \frac{r^2 - 12r}{r + 2} - \frac{28}{r + 2} \). Notice the same denominator \( r + 2 \)?
By combining the numerators over this common denominator, you get: \( \frac{r^2 - 12r - 28}{r + 2} \).

Simplifying Expressions

Simplifying an expression means making it as straightforward as possible. In math, we often do this in several steps.

• Combine like terms
• Factor where possible
• Cancel out common terms

In the given expression:

• First, we found that the common denominator allows us to combine the fractions: \( \frac{r^2 - 12r - 28}{r + 2} \).
• Then, we factored the quadratic \(r^2 - 12r - 28\) into \( (r - 14)(r + 2) \)
• Finally, we canceled out the common factor of \( r + 2 \) to get the simplified form \( r - 14 \).

Combining Fractions

Combining fractions involves merging two or more fractions into a single fraction. Here's how:

• If the denominators are the same, just add or subtract the numerators.
• If they are different, find the least common denominator first.

In our example, combining was simple because the denominators were already the same: \( r+2 \). So, we subtracted the numerators directly:

• \( \frac{r^2 - 12r}{r + 2} - \frac{28}{r + 2} = \frac{r^2 - 12r - 28}{r + 2} \)

Think of it like this:

• If you have \( \frac{a}{b} - \frac{c}{b} \), it simplifies to \( \frac{a - c}{b} \).

This way, combining fractions becomes a straightforward task.