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Chapter 7: Problem 27

For exercises \(5-48\), simplify. $$ \frac{x^{2}}{x-9}-\frac{7 x+18}{x-9} $$

Short Answer

Expert verified
The simplified form is \( x + 2 \).

Step by step solution

01

Combine the Fractions

Since both fractions have the same denominator, combine them into a single fraction: \[ \frac{x^2 - (7x + 18)}{x - 9} \].
02

Simplify the Numerator

Distribute the negative sign across the terms in the numerator: \[ x^2 - 7x - 18 \].
03

Factor the Numerator

Factor the quadratic expression in the numerator: \[ (x - 9)(x + 2) \].
04

Cancel Common Factors

Cancel the common factor in the numerator and the denominator: \[ \frac{(x - 9)(x + 2)}{x - 9} = x + 2 \].

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Key Concepts

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

Combining fractions
When simplifying algebraic fractions, the first thing to check is if the fractions share a common denominator. Sharing a common denominator means you can combine them into a single fraction.
This is similar to adding or subtracting regular fractions.
For instance, in the problem \(\frac{x^{2}}{x-9} - \frac{7 x+18}{x-9}\), both fractions have \(x-9\) as their denominator.
This allows us to combine them easily into one single fraction:
$$ \frac{x^2 - (7x + 18)}{x - 9} $$
Notice that when subtracting, we need to be careful to distribute the negative sign across each term in the second fraction.
This step simplifies our work and sets us up to simplify the expression further.
Factoring quadratics
Factoring is breaking down a complex expression into simpler ones that can be multiplied to give the original expression.
With quadratics, such as \(x^2 - 7x - 18\), we look for two numbers that multiply to the constant term (here, -18) and add up to the linear coefficient (-7).
In this example, -9 and 2 fit those requirements, giving us the factored form:
$$ x^2 - 7x - 18 = (x - 9)(x + 2) $$
Factoring helps in further simplification. Here, by factoring the numerator, we transform it into a product of simpler expressions.
This practice is essential not just for simplification but also for solving quadratic equations.
Canceling common factors
After combining and factoring, the next step is often to cancel common factors.When you have the same factor in both the numerator and denominator, you can cancel them out, simplifying the fraction further.\br> In our example, the fraction \(\frac{(x - 9)(x + 2)}{x - 9}\) has \(x - 9\) as a factor in both the numerator and the denominator.Therefore, we can cancel \(x - 9\), reducing the expression to:
$$ x + 2 $$
Canceling common factors simplifies the expression and is a crucial step in arriving at the final, simplest form of the algebraic fraction.
Always look for this opportunity as it makes your work significantly easier!

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Most popular questions from this chapter

The relationship of the diameter of a circle, \(x\), and the circumference of the circle, \(y\), is a direct variation. The diameter of a circle is \(20 \mathrm{~cm}\), and the circumference is \(62.8 \mathrm{~cm}\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this relationship. c. Find the circumference of a circle with a diameter of \(40 \mathrm{~cm}\).

The relationship of the radius of a circle, \(x\), and the circumference of the circle, \(y\), is a direct variation. The radius of a circle is \(10 \mathrm{~cm}\), and the circumference is \(62.8 \mathrm{~cm}\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this relationship. c. Find the circumference of a circle with a radius of \(20 \mathrm{~cm}\).

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If \(\$ 500\) is invested for 10 years and earns simple interest, find the annual simple interest rate needed to earn \(\$ 425\) in interest.
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