- Set the quadratic expression equal to zero to find the critical points: \(x^2 - 8x + 15 = 0\).
- Factor the expression as \((x - 3)(x - 5) = 0\) to obtain the roots \(x = 3\) and \(x = 5\).
- Determine that these roots serve as the boundaries of the interval where the inequality is satisfied.
- The critical points are \(\boxed{3, 5}\).
Explanation
- Problem Analysis
The problem requires solving the quadratic inequality \(x^2 - 8x + 15 < 0\) and identifying its critical points from a provided list. A critical point of a quadratic inequality is a value of \(x\) where the corresponding quadratic equation \(x^2 - 8x + 15 = 0\) holds true. These points act as boundaries that divide the number line into intervals where the quadratic expression is either positive or negative.
- Finding Critical Points
To find the critical points, we set the expression equal to zero:
\[x^2 - 8x + 15 = 0\]
We can factor this quadratic expression by looking for two numbers that multiply to 15 and sum to -8. These numbers are -3 and -5. Thus, the equation can be written as:
\[(x - 3)(x - 5) = 0\]
Setting each factor to zero gives us the critical points:
\[x - 3 = 0 \implies x = 3\]
\[x - 5 = 0 \implies x = 5\]
The critical points of the inequality are 3 and 5.
- Verifying Intervals and Conclusion
The critical points 3 and 5 divide the real number line into three intervals: \((-\infty, 3)\), \((3, 5)\), and \((5, \infty)\). We test a value in each interval to see where the inequality \(x^2 - 8x + 15 < 0\) holds:
- For \(x < 3\), testing \(x = 0\): \(0^2 - 8(0) + 15 = 15 > 0\).
- For \(3 < x < 5\), testing \(x = 4\): \(4^2 - 8(4) + 15 = 16 - 32 + 15 = -1 < 0\).
- For \(x > 5\), testing \(x = 6\): \(6^2 - 8(6) + 15 = 36 - 48 + 15 = 3 > 0\).
The inequality is satisfied in the interval \((3, 5)\). Comparing our critical points 3 and 5 with the given options, we confirm that 3 and 5 are the correct choices.
Examples
Understanding critical points is fundamental in fields like engineering and economics. For example, if a company models its profit \(P\) as a function of the price \(x\) of a product using a quadratic equation \(P(x) = -x^2 + 8x - 15\), finding the critical points helps determine the price range where the profit is positive (\(P(x) > 0\)). Identifying these boundaries allows businesses to set prices effectively to ensure profitability.