A train travels a distance of 720 km at a uniform speed. If the speed has been 12 km/hour less, then it would have taken 3 hours more to cover the same distance. Represent the above problem mathematically in terms of a quadratic equation.

Answer

To represent the problem mathematically in terms of a quadratic equation, we can follow these steps:

### Step 1: Define Variables
Let the speed of the train be \( V \) km/h. The distance traveled by the train is given as 720 km.

### Step 2: Establish the Time Equation
Using the formula for distance, we know:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
Thus, the time taken to cover 720 km at speed \( V \) is:
\[ t = \frac{720}{V} \]

### Step 3: Establish the Second Condition
If the speed were 12 km/h less, the speed would be \( V - 12 \) km/h. According to the problem, this would take 3 hours more time:
\[ t + 3 = \frac{720}{V - 12} \]

### Step 4: Set Up the Equation
Now we can set up the equation based on the second condition:
\[ \frac{720}{V - 12} = \frac{720}{V} + 3 \]

### Step 5: Clear the Fractions
To eliminate the fractions, we can multiply through by \( V(V - 12) \):
\[ 720V = 720(V - 12) + 3V(V - 12) \]

### Step 6: Expand and Rearrange
Expanding both sides gives:
\[ 720V = 720V - 8640 + 3V^2 - 36V \]
Now, simplify this:
\[ 0 = 3V^2 - 36V - 8640 \]

### Step 7: Divide by 3
To simplify the equation further, divide the entire equation by 3:
\[ 0 = V^2 - 12V - 2880 \]

### Step 8: Write the Quadratic Equation
Thus, the quadratic equation representing the problem is:
\[ V^2 - 12V - 2880 = 0 \]