Step 1: Understand the problem statement
The problem requires finding a value for x that makes the expression (x^2 + 3x) / (x - 4) equal to zero.The given equation is a rational function, and we need to find the values of x that make the numerator equal to zero. To do this, we can start by factoring out an x from the numerator.
Step 1: Understanding the Rational Function
| Rational Function Definition | Description |
|---|---|
| (x^2 + 3x) / (x - 4) | A rational function with a quadratic expression in the numerator and a linear expression in the denominator. |
Step 1: Properties of Rational Functions
Rational functions have several important properties that can help us solve equations like this one. One key property is that the function will be undefined when the denominator is equal to zero.
| Property | Description |
|---|---|
| The function is undefined when the denominator is zero. | This means that if we substitute x = a into the equation and get a zero in the denominator, then the function will be undefined at that value of x. |
Step 2: Set up the equation
To find the values of x that make the expression equal to zero, we can set the numerator equal to zero and solve for x.
(x^2 + 3x) / (x - 4) = 0
Step 2: Setting Up the Equation
| Equation | Description | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
(x^2 + 3x) / (x - 4)
Common Questions About Solving Rational EquationsHere are some common questions that readers may have about solving rational equations, along with detailed answers to help you quickly find the information you need. Q: What is a rational equation and how do I solve it?A rational equation is an equation in which the unknown variable is part of a rational expression, meaning it is divided by another value. To solve a rational equation, first factor out any common factors from the numerator and denominator, then cross-multiply to eliminate the fraction. Q: How do I factor out x from the numerator?To factor out x from the numerator, look for any common factors among the terms. In this case, we can factor out an x from both x^2 and 3x, resulting in x(x + 3).
Q: What happens when the denominator is zero?If the denominator is equal to zero, then the function is undefined at that value of x. In this case, we would need to find any restrictions on the domain and exclude those values from our solution set.
Q: How do I simplify a rational expression?To simplify a rational expression, factor out any common factors from the numerator and denominator, then cancel out any matching terms. This can help reduce the complexity of the expression and make it easier to work with.
Q: Can I solve rational equations with fractions?Yes, you can solve rational equations that contain fractions. To do this, first simplify the fraction by finding a common denominator, then cross-multiply to eliminate the fraction.
Q: How do I graph a rational function?To graph a rational function, first identify any vertical asymptotes or holes in the graph. Then, plot any additional points on the graph based on the simplified rational expression.
Q: Can I solve rational equations with quadratic expressions?Yes, you can solve rational equations that contain quadratic expressions. To do this, first factor out any common factors from the numerator and denominator, then cross-multiply to eliminate the fraction.
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