Problem 27
Question
For exercises \(5-48\), simplify. $$ \frac{x^{2}}{x-9}-\frac{7 x+18}{x-9} $$
Step-by-Step Solution
Verified Answer
The simplified form is \( x + 2 \).
1Step 1: Combine the Fractions
Since both fractions have the same denominator, combine them into a single fraction: \[ \frac{x^2 - (7x + 18)}{x - 9} \].
2Step 2: Simplify the Numerator
Distribute the negative sign across the terms in the numerator: \[ x^2 - 7x - 18 \].
3Step 3: Factor the Numerator
Factor the quadratic expression in the numerator: \[ (x - 9)(x + 2) \].
4Step 4: Cancel Common Factors
Cancel the common factor in the numerator and the denominator: \[ \frac{(x - 9)(x + 2)}{x - 9} = x + 2 \].
Key Concepts
Combining fractionsFactoring quadraticsCanceling common factors
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.
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.
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!
$$ x + 2 $$
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!
Other exercises in this chapter
Problem 27
For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{2}}{\frac{1}{3}+\frac{1}{5}} $$
View solution Problem 27
For exercises 27-34, evaluate. $$ \frac{1}{12}+\frac{5}{12} $$
View solution Problem 27
For exercises 7-32, simplify. $$ \frac{2 x^{2}-5 x-3}{x^{2}-12 x+27} \cdot \frac{x^{2}-15 x+54}{2 x^{2}+13 x+6} $$
View solution Problem 27
For exercises 1-66, simplify. $$ \frac{6 x+6}{3 x-3} $$
View solution