Answer :

Let the speed of the train be x km/hr

The total journey distance, d = 300km….(i)

Case 1: with usual speed

usual speed of the train, s = x km/hr

We know Distance = speed × time

So usual time,

Case 2: with increased speed

new speed of the train, s_{1} = (x + 5) km/hr

We know Distance = speed × time

So increases time,

Now it is given the train takes 2 hours less after increase in speed

So increased time = original time - 2 hours

i.e., t_{1} = t - 2

Substituting values from equation (ii) and (iii), we get

⇒ 300x = (x + 5) (300 - 2x)

⇒ 300x = x(300 - 2x) + 5(300 - 2x)

Cancelling the like terms, we get

⇒ 300x = 300x - 2x^{2} + 1500 - 10x

⇒ 2x^{2} + 10x - 1500 = 0

Dividing throughout by 2, we get

⇒ x^{2} + 5x - 750 = 0

Now we will be factorizing this by splitting the middle term, we get

⇒ x^{2} + 30x - 25x - 750 = 0

⇒ x (x + 30) - 25(x + 30) = 0

⇒ (x + 30) (x - 25) = 0

⇒ (x + 30) = 0 or (x - 25) = 0

⇒ x = - 30 or x = 25

But x is the speed of the train and it cannot be negative, hence the usual speed of the train is 25km/hr.

**OR**

Cancelling the like terms, we get

⇒ - ab = x (a + b + x)

⇒ - ab = ax + bx + x^{2}

⇒ x^{2} + (a + b) x + ab = 0

Comparing this with standard equation, i.e., Ax^{2} + Bx + C = 0, we get

A = 1, B = (a + b), C = ab

So, the value of x of a quadratic equation can be found by using the formula,

Now substituting the corresponding values, we get

But we know the expansion of (a + b)^{2} = a^{2} + b^{2} + 2ab, substituting this in above equation, we get

But we know a^{2} + b^{2} - 2ab = (a - b)^{2}, substituting this in above equation, we get

Now this has two possibilities,

⇒ x = - b or x = - a

These are the required values of x.

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