Can't get a proper answer using concepts of quadratics

Question

Take a look at the problem:

Amirah travels by coach from Singapore to Penang, which are 700 km apart, to visit her grandparents. she returns to singapore by car at an average speed which is 30 km/h greater than that of a coach. If the average speed of the car is $x$ km/h and the time taken for the whole journey is $20 $ hours, formulae an equation in $x $ and show that it reduces to $x^2 - 100x + 1050 = 0$.

This is what I managed to do so far and I generated $3$ equations but the answer isn't correct:

1. $x + 30$
2. $700/x$
3. $700/(x + 30)$

$$700/x = 700/(x + 30) - 20$$ The equation produced by this is $x^2 + 30x + 1050 = 0$.

Where am I making a mistake?

Answer 1

Average speed throughout the journey is $$\frac{700 ~\mathrm{km}}{20 ~\mathrm{h}} = 35 ~\mathrm{km/h}.$$

But the same average speed is given by the harmonic average of the speeds of the 2 legs, hence $$ 35 = \frac2{\frac1x + \frac1{x-30}} = \frac{2x(x-30)}{2x-30}, $$ can you take it from here?

Answer 2

If we take $x$ as the speed of the car and $x-30$ as the speed of the coach then we get the equation as $\dfrac{700}{x}+\dfrac{700}{x-30}=20$$$700(x-30)+700x=20x(x-30)$$$$700x-21000+700x=20x^2-600x$$$$20x^2-2000x+21000=0$$$$20(x^2-100x+1050)=0$$$$x^2-100x+1050=0$$

Answer 3

You have the right quantities and expressions in your work, but the way you combined them is indeed incorrect. It would help a lot (to yourself!) if you label each quantity and each expression and each equation that you set up.

For example, what does $\frac{700}{x}$ represent? Dividing distance by average speed gives us the time traveled, so apparently $\frac{700}{x}$ is the time of traveling by car, i.e. the time (duration) of the trip back. Similarly, $\frac{700}{x-30}$ represents the time (duration) of the trip by coach.

Now, your equation $$\frac{700}{x}=\frac{700}{x-30}-20,$$ when translated to words, says that if we take the duration of the ride by coach and subtract 20 hours, we'd get the duration of the ride by car. In other words, it says that the ride by car was 20 hours shorter than the ride by coach. Is that what the text says? I'm afraid, no.

The correct equation will have the same three quantities. But make sure that you arrange them into an equation that says the same thing as the problem says.