24. This holy week, Abe went up to Baguio City in his motorcycle at an average rate of 30 Kilometer per hour. Going downhill, he returned at a faster rate of 45 KPH. What is his average speed for the entire trip in KPH?

3 points

35

36

37.5

38

39

25. What is the coefficient of the x^2 term in the product of (2x^2 + 3x - 4) and (3x^2 - 4x + 6)?

2 points

24

12

0

- 12

- 24

### Problem 24: Average Speed Calculation
Abe goes uphill at 30 KPH and downhill at 45 KPH. To find the average speed for the entire trip, we use the formula for average speed in round trips, which is:

\[
\text{Average Speed} = \frac{2 \cdot v_1 \cdot v_2}{v_1 + v_2}
\]

where:
- \(v_1 = 30 \, \text{KPH}\) (uphill speed)
- \(v_2 = 45 \, \text{KPH}\) (downhill speed)

Substitute the values into the formula:

\[
\text{Average Speed} = \frac{2 \cdot 30 \cdot 45}{30 + 45} = \frac{2 \cdot 1350}{75} = \frac{2700}{75} = 36 \, \text{KPH}
\]

So, the average speed for the entire trip is **36 KPH**.

### Answer for Problem 24: **36 KPH**

---

### Problem 25: Coefficient of \(x^2\) Term
We need to find the coefficient of the \(x^2\) term in the product of \((2x^2 + 3x - 4)\) and \((3x^2 - 4x + 6)\).

Expand the expression by multiplying each term:

\[
(2x^2 + 3x - 4)(3x^2 - 4x + 6)
\]

Now, focus only on the terms that will result in an \(x^2\) term when multiplied:
- \(2x^2 \cdot 6 = 12x^2\)
- \(3x \cdot (-4x) = -12x^2\)
- \((-4) \cdot (3x^2) = -12x^2\)

Now sum the \(x^2\) terms:

\[
12x^2 - 12x^2 - 12x^2 = -12x^2
\]

Thus, the coefficient of the \(x^2\) term is **-12**.

### Answer for Problem 25: **-12**

---

Would you like more details or have any questions about the solutions?

Here are 5 related questions for further practice:

1. If Abe had taken 2 hours to travel uphill and 1.5 hours to travel downhill, what would be his total distance traveled?
2. What is the average speed if Abe's uphill speed is increased to 40 KPH and his downhill speed remains 45 KPH?
3. How many terms are there in the expanded form of \((2x^2 + 3x - 4)(3x^2 - 4x + 6)\)?
4. What is the constant term in the expansion of \((2x^2 + 3x - 4)(3x^2 - 4x + 6)\)?
5. What is the sum of the coefficients of the expanded polynomial \((2x^2 + 3x - 4)(3x^2 - 4x + 6)\)?

### Tip:
For round trips with different speeds, always use the harmonic mean formula for average speed instead of simply averaging the two speeds directly.

24. This holy week, Abe went up to Baguio City in his motorcycle at an average rate of 30 Kilometer per hour. Going downhill, he returned at a faster rate of 45 KPH. What is his average speed for the entire trip in KPH?

25. What is the coefficient of the x^2 term in the product of (2x^2 + 3x - 4) and (3x^2 - 4x + 6)?

Solve two math problems: find Abe's average speed for a round trip to Baguio at speeds of 30 KPH uphill and 45 KPH downhill, and determine the coefficient of the x^2 term in the product of two polynomials (2x^2 + 3x - 4) and (3x^2 - 4x + 6). Learn to apply harmonic mean for speed and the distributive property in polynomial multiplication.

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