The average speed is calculated using the formula:

$\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}$

Given:

• Distance covered = $15x^2 + 7x - 2$ km
• Time taken = $3x - 2$ hours

So, the average speed is:

$\text{Average speed} = \frac{15x^2 + 7x - 2}{3x - 2}$

This is a rational expression, so no further simplification is possible unless the numerator and denominator have common factors. Let's check that:

Factorizing the numerator $15x^2 + 7x - 2$, we get:

• Factors of $15x^2$ and $-2$ that combine to give $7x$ are tricky to find, so we'll leave it as is for now.

Thus, the expression for the average speed remains:

$\boxed{\frac{15x^2 + 7x - 2}{3x - 2}} \, \text{km/hr}$

Would you like further details on factorization or simplification methods?

Related Questions:

1. How would the result change if the distance expression had different coefficients?
2. What happens to the average speed as $x$ approaches certain values, like $\frac{2}{3}$?
3. Can you express the average speed for a specific value of $x$?
4. What is the physical meaning of each term in the quadratic expression for distance?
5. How can you graph the average speed for varying values of $x$?

Tip:

When simplifying rational expressions, always check for common factors to potentially reduce the expression.