A train travels a distance of 480 km at a uniform speed. If the speed had been $8~km/h$ less, then it would have taken 3 hours more to cover the same distance. Formulate the quadratic equation in terms of the speed of the train

Question

Accepted Answer

To solve this problem, we need to formulate a quadratic equation based on the given conditions about the train's speed and travel time.

Let's denote the original speed of the train as $ x $ km/h. According to the problem, the train covers a distance of 480 km at this speed. Therefore, the time taken to cover this distance is given by:

$$
\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{480}{x} \text{ hours}
$$

If the speed had been $ 8 $ km/h less, the new speed would be $ x - 8 $ km/h. The time taken to cover the same distance at this reduced speed would be:

$$
\text{New Time} = \frac{480}{x - 8} \text{ hours}
$$

According to the problem, this new time is 3 hours more than the original time. Therefore, we can set up the equation:

$$
\frac{480}{x - 8} = \frac{480}{x} + 3
$$

To eliminate the fractions, we multiply through by $ x(x - 8) $:

$$
480x = 480(x - 8) + 3x(x - 8)
$$

Expanding and simplifying the equation:

$$
480x = 480x - 3840 + 3x^2 - 24x
$$

Cancel $ 480x $ from both sides:

$$
0 = 3x^2 - 24x - 3840
$$

Divide the entire equation by 3 to simplify:

$$
0 = x^2 - 8x - 1280
$$

This is the quadratic equation we were asked to formulate. It matches Option 3: $ x^2 - 8x - 1280 = 0 $.

Now, let's explain why the other options are incorrect:

1. **Option 1: $ x^2 + 8x + 1280 = 0 $**

This equation has a positive $ 8x $ term and a positive constant term, which does not match the derived equation. The problem specifies a decrease in speed, leading to a negative $ 8x $ term and a negative constant when rearranged.

2. **Option 2: $ x^2 - 8x + 1280 = 0 $**

This equation has a positive constant term, which does not align with the derived equation. The constant term should be negative because the equation results from subtracting a larger time from a smaller time, leading to a negative result.

3. **Option 4: $ x^2 + 8x - 1280 = 0 $**

This equation has a positive $ 8x $ term, which is incorrect. The problem involves a reduction in speed, resulting in a negative $ 8x $ term.

Therefore, Option 3 is the correct formulation of the quadratic equation based on the given problem conditions.