Used car breaks down

The warranty is irrelevant in this case as you would be covered by your statutory rights under Sale of Goods - which states that items purchased from a commercial entity should be of reasonable quality and durable, taking into account factors such as consumer expectations and price paid. In this case the price you paid was significant and you would have expected the car to be roadworthy for a greater length of time. Therefore you have sound grounds to argue that the dealership are in breach of Sale of Goods Section 14. Your recourse is with the company who sold you the car and any refusal to accept responsibility should be referred to either Trading Standards or a Small Claims Court. For further info see the section on consumer rights (buying goods) here: They say I've taken ownership so it's my responsibility | Consumer Information
and here: Second hand goods and your rights | Consumer Information

From my experience it is not uncommon for businesses to say no to legitimate claims such as this. They know that most people will just accept it and being unsure of their rights will just walk away.

Tony