On SpaceX's website, they make a claim that their rockets are the safest and most reliable in the industry. This is because apparently it can complete its mission even if up-to two of its engines fail during flight.
How is this possible? Wouldn't the lack of the engine thrust from two engines cause thrust imbalance and throw the rocket off course or destabilize it?
The Falcon 9 can shut off a faulty engine, re-configure the remaining engines (change thrust levels, vector direction) and update its flight profile in real-time. This happened on the SpaceX CRS-1 flight (see also this article).
The Merlin engines can be gimbaled (change the direction of thrust), so on the Falcon 9 all engines can slightly change the thrust direction. This can be used to offset the lack of thrust from the failed engine.
Another way the Falcon 9 could compensate the thrust imbalance would be to switch off the engine opposite of the failed engine. That's what the infamous N-1 tried to do… but failed to implement properly.
The engines can be vectored to manage this. They actually have had an engine failure and still made it to orbit in the CRS-1 mission. Note that there are some points in the mission that an engine failure is more likely to lead to a mission failure than others. It should be noted that while a 1 engine failure has been recovered from, a 2 engine failure has not been demonstrated, and would be significantly more difficult to recover from.
Most notably among these is the upper stage cannot fail. This is less likely than one of the lower stage failing, because it doesn't have to interact with other engines, but is still a risk.
Per the specs on the F9 page it has about a 1.4 to 1 thrust to weight ratio at takeoff and it goes up from there as the vehicle burns fuel getting lighter and the engines start to generate more thrust in the higher atmosphere. So theoretically if two engines failed right after lift off it would be down to ~1.1 thrust to weight ratio so it can still accelerate upwards and as others have mentioned the thrust can be vectored to compensate for the unbalanced thrust. Now it may not be able to handle any two engine out scenario so two engines right next to each other that stop right after take off might be too much. It might not be able to vector thrust enough to compensate while still keeping the thrust to weight ratio up enough to actually launch.
Another factor that hasn't really been mentioned is fuel reserves. Loosing an engine will mean lower performance which means with the same amount of fuel the first stage won't get the second stage going as fast because you spend more time fighting gravity. Normally this would mean the payload could make it to orbit but might not be in it's intended orbit. That would typically mean the payload would have to use more of it's own fuel to get to the correct orbit and would shorten it's useful life. The F9 for most launches has a pretty large reserve intended for landing the rocket but that could be used to ensure the payload gets to the intended orbit. Now even expendable launches typically have reserves that can be used to help cover for that and on some launches have even been used to be launched into super-synchronous orbits beyond what was required which ends up extending the life of the payload.
