A car travels a distance of 72 km at a certain average speed of x km per hour and then travels a distance of 81 km at an average speed of 6 km per hour more than its original average speed. If it takes 3 hours to complete the total journey then form a quadratic equation and solve it to find its original average speed.

Given,

A car travels a distance of 72 km at a certain average speed of x km per hour and then travels a distance of 81 km at an average speed of 6 km per hour more than its original average speed.

Total time taken to complete the journey = 3 hours

$\therefore \dfrac{72}{x} + \dfrac{81}{x + 6} = 3 \\[1em] \Rightarrow \dfrac{72(x + 6) + 81x}{x(x + 6)} = 3 \\[1em] \Rightarrow 72(x + 6) + 81x = 3x(x + 6) \\[1em] \Rightarrow 72x + 432 + 81x = 3x^2 + 18x \\[1em] \Rightarrow 3x^2 + 18x - 72x - 81x - 432 = 0 \\[1em] \Rightarrow 3x^2 - 135x - 432 = 0 \\[1em] \Rightarrow 3(x^2 - 45x - 144) = 0 \\[1em] \Rightarrow x^2 - 45x - 144 = 0 \\[1em] \Rightarrow x^2 - 48x + 3x - 144 = 0 \\[1em] \Rightarrow x(x - 48) + 3(x - 48) = 0 \\[1em] \Rightarrow (x + 3)(x - 48) = 0 \\[1em] \Rightarrow x + 3 = 0 \text{ or } x - 48 = 0 \\[1em] \Rightarrow x = -3 \text{ or } x = 48.$

As speed cannot be negative in this case,

Hence, the original speed = 48 km/hr.