An analyst has found that his​ company's profits, in hundreds of thousands of​ dollars, are given by the following​ equation, where x is the​ amount, in hundreds of​ dollars, spent on advertising. For what values of x does the company make a​ profit?
Pequals3xsquaredminus17xplus20
Question content area bottom
Part 1
For what values of x does the company make a​ profit?
  

​(Type your answer in interval notation. Use integers or fractions for any numbers in the​ expression.)

The equation for the company's profits is:

\[
P(x) = 3x^2 - 17x + 20
\]

To find when the company makes a profit, we need to determine the values of \(x\) where \(P(x) > 0\). This is a quadratic inequality.

### Step 1: Find the roots of the quadratic equation
First, solve for \(P(x) = 0\) to find the critical points.

\[
3x^2 - 17x + 20 = 0
\]

Use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For \(3x^2 - 17x + 20\), \(a = 3\), \(b = -17\), and \(c = 20\). Plug these into the quadratic formula:

\[
x = \frac{-(-17) \pm \sqrt{(-17)^2 - 4(3)(20)}}{2(3)}
\]
\[
x = \frac{17 \pm \sqrt{289 - 240}}{6}
\]
\[
x = \frac{17 \pm \sqrt{49}}{6}
\]
\[
x = \frac{17 \pm 7}{6}
\]

Thus, the roots are:

\[
x = \frac{17 + 7}{6} = 4 \quad \text{and} \quad x = \frac{17 - 7}{6} = \frac{10}{6} = \frac{5}{3}
\]

### Step 2: Analyze the intervals
The roots split the number line into three intervals: \( (-\infty, \frac{5}{3}) \), \( \left(\frac{5}{3}, 4\right) \), and \( (4, \infty) \).

We now test a point from each interval in the profit function \(P(x)\) to see whether \(P(x)\) is positive or negative.

- For \(x = 0\) (in the interval \((-\infty, \frac{5}{3})\)):  
  \[
  P(0) = 3(0)^2 - 17(0) + 20 = 20 \quad (\text{positive})
  \]
- For \(x = 2\) (in the interval \((\frac{5}{3}, 4)\)):  
  \[
  P(2) = 3(2)^2 - 17(2) + 20 = 3(4) - 34 + 20 = 12 - 34 + 20 = -2 \quad (\text{negative})
  \]
- For \(x = 5\) (in the interval \((4, \infty)\)):  
  \[
  P(5) = 3(5)^2 - 17(5) + 20 = 3(25) - 85 + 20 = 75 - 85 + 20 = 10 \quad (\text{positive})
  \]

### Step 3: Conclusion
The company makes a profit when \(P(x) > 0\), which occurs in the intervals where the function is positive.

Thus, the company makes a profit when \(x \in (-\infty, \frac{5}{3}) \cup (4, \infty)\).

### Final Answer:
\[
x \in (-\infty, \frac{5}{3}) \cup (4, \infty)
\]

Would you like further details on this solution?

### Related Questions:
1. How does the discriminant affect the nature of the roots in a quadratic equation?
2. What is interval notation, and how is it used in inequalities?
3. Can a quadratic equation have no real solutions? If so, what does that mean for the company in this scenario?
4. How would the profit function change if the coefficients were altered?
5. Can you explain how to graph quadratic inequalities?

### Tip:
When solving quadratic inequalities, always check intervals around the critical points to determine where the function is positive or negative.

An analyst has found that his​ company's profits, in hundreds of thousands of​ dollars, are given by the following​ equation, where x is the​ amount, in hundreds of​ dollars, spent on advertising. For what values of x does the company make a​ profit?
P(x) = 3x^2 - 17x + 20.
For what values of x does the company make a​ profit?
(Type your answer in interval notation. Use integers or fractions for any numbers in the​ expression.)

Discover how to solve quadratic inequalities to determine when a company's profit function is positive based on the advertising spending equation P(x) = 3x^2 - 17x + 20. Explore key concepts in algebra and quadratic equations, with step-by-step instructions. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation