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Chapter 7: Problem 15

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

Short Answer

Expert verified
\( r - 14 \)

Step by step solution

01

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.
02

Combine the fractions

Combine the fractions using the common denominator: \(\frac{r^2 - 12r}{r+2} - \frac{28}{r+2} = \frac{r^2 - 12r - 28}{r+2}\).
03

Simplify the numerator

Factorize the quadratic expression in the numerator. Note that \(r^2 - 12r - 28\) can be factored as follows: \( (r - 14)(r + 2)\).
04

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.
05

Write the final simplified expression

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

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