Problem 85
Question
For exercises 77-86, find any values of the variable for which this expression is undefined. $$ \frac{2}{x^{2}+3} $$
Step-by-Step Solution
Verified Answer
There are no real values of \(x\) that make the expression undefined.
1Step 1: Understand when a fraction is undefined
A fraction is undefined when its denominator is equal to 0. In this case, we have the denominator as \(x^2 + 3\). Hence, we need to find the values for which \(x^2 + 3 = 0\).
2Step 2: Set the denominator equal to zero
Set the equation to zero: \(x^2 + 3 = 0\).
3Step 3: Solve for the variable
Subtract 3 from both sides to isolate \(x^2\): \(x^2 = -3\).
4Step 4: Analyze the result
Realize that there are no real numbers that satisfy \(x^2 = -3\), as the square of a real number is always non-negative. Therefore, there are no real values of \(x\) that make the expression undefined.
Key Concepts
Understanding FractionsWhat are Undefined Expressions?Focusing on the Denominator
Understanding Fractions
Before delving into undefined fractions, it's essential to understand what fractions are. A fraction represents a part of a whole or any number of equal parts. It consists of a numerator (the top part) and a denominator (the bottom part). The numerator indicates how many parts you have, while the denominator shows how many equal parts the whole is divided into.
For instance, in the fraction \(\frac{2}{3}\), 2 is the numerator, and 3 is the denominator. This means we have 2 parts out of a total of 3 equal parts.
In our exercise, the fraction is \(\frac{2}{x^2 + 3}\). The key part here is the denominator, which will help us determine if the fraction is undefined.
For instance, in the fraction \(\frac{2}{3}\), 2 is the numerator, and 3 is the denominator. This means we have 2 parts out of a total of 3 equal parts.
In our exercise, the fraction is \(\frac{2}{x^2 + 3}\). The key part here is the denominator, which will help us determine if the fraction is undefined.
What are Undefined Expressions?
An expression is defined by the values it can take. Sometimes, certain values lead to the expression being undefined.
In the context of fractions, an undefined fraction occurs when the denominator is zero. This is because division by zero is not allowed in mathematics, as it doesn't produce a meaningful result.
For example, if we take the fraction \(\frac{a}{b}\) and set the denominator \(b = 0\), we'd end up with \(\frac{a}{0}\), which is undefined. This is exactly what we are looking for in our exercise. We set the denominator equal to zero and solve for the variable.
In the context of fractions, an undefined fraction occurs when the denominator is zero. This is because division by zero is not allowed in mathematics, as it doesn't produce a meaningful result.
For example, if we take the fraction \(\frac{a}{b}\) and set the denominator \(b = 0\), we'd end up with \(\frac{a}{0}\), which is undefined. This is exactly what we are looking for in our exercise. We set the denominator equal to zero and solve for the variable.
Focusing on the Denominator
The denominator is a crucial part of evaluating whether a fraction is defined or not. Let's take a closer look at the denominator in our given fraction, which is \(x^2 + 3\).
To determine if there are any values of \(x\) that make the fraction undefined, we set the denominator equal to zero: \(x^2 + 3 = 0\).
Solving for \(x\), we get: \[ x^2 = -3. \]
Here, we see a key point: the square of any real number is always non-negative. This means there are no real numbers that satisfy \(x^2 = -3\). Therefore, there are no values of \(x\) that make the expression \(\frac{2}{x^2 + 3}\) undefined.
Understanding how to manipulate the denominator to find undefined expressions is essential to mastering algebraic fractions.
To determine if there are any values of \(x\) that make the fraction undefined, we set the denominator equal to zero: \(x^2 + 3 = 0\).
Solving for \(x\), we get: \[ x^2 = -3. \]
Here, we see a key point: the square of any real number is always non-negative. This means there are no real numbers that satisfy \(x^2 = -3\). Therefore, there are no values of \(x\) that make the expression \(\frac{2}{x^2 + 3}\) undefined.
Understanding how to manipulate the denominator to find undefined expressions is essential to mastering algebraic fractions.
Other exercises in this chapter
Problem 84
For exercises \(35-86\), simplify. $$ \frac{1}{6 n}+\frac{3}{2 n}-\frac{7}{4 n} $$
View solution Problem 85
For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 10 x^{4}+20 x^{3}+10 x^{2} ; 8 x^{3}+16 x^{2}+8 x $$
View solution Problem 86
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View solution Problem 86
For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 6 x^{4}+24 x^{3}+24 x^{2} ; 9 x^{3}+36 x^{2}+36 x $$
View solution