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The inequality given is:

\[
x^2 - 8x + 15 < 0
\]

To solve this quadratic inequality, let's first solve the corresponding quadratic equation:

\[
x^2 - 8x + 15 = 0
\]

Factoring the quadratic:

\[
(x - 3)(x - 5) = 0
\]

So, the solutions to the equation are \( x = 3 \) and \( x = 5 \).

Now, to solve the inequality \( (x - 3)(x - 5) < 0 \), we analyze the sign of the product in the intervals determined by the roots \( x = 3 \) and \( x = 5 \):

1. For \( x < 3 \), both factors \( (x - 3) \) and \( (x - 5) \) are negative, so their product is positive.
2. For \( 3 < x < 5 \), \( (x - 3) \) is positive and \( (x - 5) \) is negative, so their product is negative.
3. For \( x > 5 \), both factors are positive, so their product is positive.

Therefore, the inequality is satisfied when \( 3 < x < 5 \).

The solution is:

\[
x \in (3, 5)
\]

Would you like more details on any part of this solution?

### Related Questions:
1. What are the methods to solve quadratic inequalities?
2. How do we determine signs of quadratic expressions across intervals?
3. Why does factoring help in solving inequalities?
4. What happens when an inequality has no real solutions?
5. Can you solve a quadratic inequality using the quadratic formula?

### Tip:
Always check critical points (like roots of the equation) when solving inequalities by factoring.

Solve the quadratic inequality: x^2 - 8x + 15 < 0

Learn how to solve the quadratic inequality x^2 - 8x + 15 < 0 by factoring and analyzing the intervals. This step-by-step solution helps students understand key concepts in solving inequalities.

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