Q2(iv):

Represent the following situations in the form of quadratic equations : (iv) 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. We need to find the speed of the train.

Given:

• Total distance to be covered by the train, $d = 480$ km.
• The train travels at a uniform speed.
• If the speed were $8$ km/h less, the time taken to cover the same distance would increase by $3$ hours.

To Find:

Represent the given situation in the form of a quadratic equation in terms of the speed of the train.

Step 1: Defining Variables

Let the uniform speed of the train be $x$ km/h.

We know the fundamental relationship between distance, speed, and time is given by the formula:

$\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

Step 2: Formulating Time Expressions

Case 1: When the train travels at the original uniform speed $x$ km/h.

Time taken ($t_1$) = $\frac{480}{x}$ hours.

Case 2: When the speed is reduced by $8$ km/h.

New speed = $(x - 8)$ km/h.

Time taken ($t_2$) = $\frac{480}{x - 8}$ hours.

Step 3: Establishing the Relationship

According to the problem, the train takes $3$ hours more when the speed is reduced. Therefore, the difference between the new time ($t_2$) and the original time ($t_1$) is $3$ hours:

$t_2 - t_1 = 3$

Substituting the expressions from Step 2:

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

Step 4: Simplifying the Equation

Divide the entire equation by $3$ to simplify the coefficients:

$\frac{160}{x - 8} - \frac{160}{x} = 1$

Find a common denominator for the left-hand side, which is $x(x - 8)$:

$\frac{160x - 160(x - 8)}{x(x - 8)} = 1$

Expand the numerator:

$\frac{160x - 160x + 1280}{x^2 - 8x} = 1$

Simplify the numerator ($160x - 160x = 0$):

$\frac{1280}{x^2 - 8x} = 1$

Step 5: Final Algebraic Form

Multiply both sides by $(x^2 - 8x)$:

$1280 = x^2 - 8x$

Rearrange the terms to the standard form of a quadratic equation $ax^2 + bx + c = 0$:

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

Final Answer: The quadratic equation representing the situation is $x^2 - 8x - 1280 = 0$, where $x$ is the speed of the train in km/h.