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Chapter 7: Problem 49

Add or subtract as indicated. Simplify the result, if possible. $$\frac{2 x+9}{x^{2}-7 x+12}-\frac{2}{x-3}$$

Short Answer

Expert verified
\(\frac{17}{(x-4)(x-3)} \)

Step by step solution

01

Factor the Denominator

The first step is to factor the denominator of the first fraction. So, \( x^{2}-7x+12 \) becomes \( (x-4)(x-3) \). The expression becomes \( \frac{2x+9}{(x-4)(x-3)}-\frac{2}{x-3} \)
02

Find a common denominator

To add or subtract fractions, the denominators must be the same. In this case, the first fraction's denominator is \( (x-4)(x-3) \) and the second fraction's denominator is \( x-3 \). So, the second fraction is multiplied by \( (x-4)/(x-4) \) to get a common denominator. It has become \( \frac{2x+9}{(x-4)(x-3)}-\frac{2(x-4)}{(x-4)(x-3)} \)
03

Subtract the fractions

Now that there is a common denominator, the fractions can be subtracted. The result is \( \frac{2x + 9 - 2(x-4)}{(x-4)(x-3)} \)
04

Simplify the rational expression

Simplify the expression in the numerator to get the final simplified result \( \frac{2x + 9 - 2x + 8}{(x-4)(x-3)} \)After simplification, the final result is \( \frac{17}{(x-4)(x-3)} \)

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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 an essential skill in algebra that allows you to rewrite quadratic expressions as products of binomials. This is particularly useful when you need to simplify rational expressions or solve quadratic equations.
To factor a quadratic expression like \( x^2 - 7x + 12 \), you need to find two numbers that multiply to the constant term (12) and add up to the coefficient of the linear term (-7). Here, these numbers are -3 and -4. These numbers allow you to rewrite the quadratic as:
  • \( x^2 - 7x + 12 = (x-4)(x-3) \)
Factoring quadratics transforms a complex expression into a simpler product of factors, making further operations, like finding common denominators, possible.
Common Denominator
When dealing with fractions, having a common denominator is crucial for adding or subtracting them. A common denominator means both fractions share the same base, allowing you to directly combine the numerators.
In our example, the first fraction has a denominator \( (x-4)(x-3) \) while the second is \( x-3 \). To make these denominators the same, multiply the numerator and denominator of the second fraction by \( (x-4) \). This process of creating a common denominator ensures both fractions can be rewritten with \( (x-4)(x-3) \) as a shared base:
  • From \( \frac{2}{x-3} \) to \( \frac{2(x-4)}{(x-4)(x-3)} \)
With a common base established, the fractions are ready to be subtracted.
Subtracting Fractions
After aligning the denominators, subtracting fractions becomes straightforward. The operation is performed on the numerators while retaining the common denominator.
In our exercise, once the fractions are aligned under the common base \( (x-4)(x-3) \), subtract the second fraction's numerator from the first:
  • \( (2x + 9) - 2(x-4) \)
Distribute the \( -2 \) across \( (x-4) \) and subtract it from the first numerator. This gives:
  • \( 2x + 9 - 2x + 8 \)
Subtracting correctly ensures the accuracy of the next simplification step.
Simplifying Algebraic Expressions
Simplifying algebraic expressions involves combining like terms and reducing the expression to its simplest form. It's a key step to making an expression more manageable and comprehensible.
In the expression \( 2x + 9 - 2x + 8 \), you combine the like terms in the numerator. Here, the \( 2x \) and \( -2x \) cancel each other out completely, while \( 9 + 8 \) simplifies to 17:
  • Final simplified numerator: 17
Thus, the original rational expression simplifies to \( \frac{17}{(x-4)(x-3)} \). This cleaner format is easier to interpret and work with, providing clarity in algebraic calculations.

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Most popular questions from this chapter

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