A Train Travels 360 km at a Uniform Speed
Question:
A train travels 360 km at a uniform speed. If the speed had been 5 km/hr more, it would have taken 1 hour less for the same journey. Form the quadratic equation to find the speed of the train.
Solution
Let the speed of the train be
\[ x \text{ km/hr} \]
Then the increased speed is
\[ (x+5)\text{ km/hr} \]
Time taken at speed \(x\):
\[ \frac{360}{x}\text{ hours} \]
Time taken at speed \(x+5\):
\[ \frac{360}{x+5}\text{ hours} \]
Given that the journey would take 1 hour less at the higher speed,
\[ \frac{360}{x}-\frac{360}{x+5}=1 \]
Multiplying both sides by \(x(x+5)\),
\[ 360(x+5)-360x=x(x+5) \]
\[ 1800=x^2+5x \]
Bringing all terms to one side,
\[ x^2+5x-1800=0 \]
Required Quadratic Equation
\[ \boxed{x^2+5x-1800=0} \]
Answer
If \(x\) denotes the speed of the train, then the required quadratic equation is
\[ \boxed{x^2+5x-1800=0} \]
This equation can be solved to find the speed of the train.