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Answer :

Let average speed of passenger train = x km/h

Let average speed of express train = (x + 11) km/h

Time taken by passenger train to cover 132 km = \({{132} \over {x}}\) hours

Time taken by express train to cover 132 km = \({{132} \over {x + 11}}\) hours

According to the given condition,

\(\Rightarrow {{132} \over {x}} = {{132} \over {x + 11}} + 1\)

\(\Rightarrow 132({{1} \over {x}} - {{1} \over {x + 11}}) = 1\)

\(\Rightarrow 132({{x + 11 - x} \over {x(x + 11)}}) = 1\)

\(\Rightarrow 132 (11) = x (x + 11)\)

\(\Rightarrow 1452 = x^2 + 11x\)

\(\Rightarrow x^2 + 11x -1452 = 0\)

Comparing equation \(x^2 + 11x -1452 = 0\) with general quadratic equation \(ax^2 + bx + c = 0\), we get a = 1, b = 11 and c = -1452

Applying quadratic formula = \(x = {{-b ± \sqrt{b^2 - 4ac}} \over {2a}}\)

\(\Rightarrow x = {{-11 ± \sqrt{(11)^2 - 4(1)(-1452)}} \over {(2)(1)}}\)

\(\Rightarrow x = {{-11 ± \sqrt{121 + 5808}} \over {2}} = {{-11 ± \sqrt{5929}} \over {2}}\)

\(\Rightarrow x = {{-11 + 77} \over {2}} , {{-11 - 77} \over {2}}\)

\(\Rightarrow x = 33, -44\)

As speed cannot be in negative. Therefore, speed of passenger train = 33 km/h

And, speed of express train = x + 11 = 33 + 11 = 44 km/h

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