Killing dandelions in lawn

Our lawn in overrun with dandelion plants. I'm talking dozens of them! Some are young, some much larger/older.
I'd like to remove them, particularly in an area where I want to introduce some spring bulbs (snowdrops, daffodils etc). I'd like to get the new bulbs in the ground in Autumn ready for next spring, but need to tackle these dandelions first.
What's the best solution? I'm torn between using a weeding tool that would (hopefully) pull the root out (like this one), or going with the weed killer solution.
Thanks in advance!
I'd like to remove them, particularly in an area where I want to introduce some spring bulbs (snowdrops, daffodils etc). I'd like to get the new bulbs in the ground in Autumn ready for next spring, but need to tackle these dandelions first.
What's the best solution? I'm torn between using a weeding tool that would (hopefully) pull the root out (like this one), or going with the weed killer solution.
Thanks in advance!

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Not sure you can actually buy one at the moment. I wouldn't know where to go as most shops are closed here
B'n'Q sell it for £5 more I just checked
@josusa47 Completely get that they don't ever really go away. The problem is the grass area is literally covered in them making it impossible to plant anything else there. Plus the more I have in the garden, the more the flowers will grow and spread seed potentially into my meadow area and borders which I want to avoid. I'll just have to keep on top of it!
Once cleared then it would be easier to dig out any new ones.
Or if you have time on your hands .....
I am unsure whether the vinegar option would kill the roots as well and if it's a big area you would need a huge amount and a pump sprayer would be best to save your hands.
If the ground's damp enough, apply a weed and feed product. That will get the bulk of them away. You can then dig out or spot kill.
I did that on moving here 7 years ago. The grass was non existent, and mainly moss, dandelions and buttercups. I now don't use anything on it, other than the odd bit of lawn weedkiller fro the dandelions which blow in from across the road.
An even better system is to use a feed only, let everything grow a bit and then use a weed and feed. The more foliage it has to work on, the more successful the weed killing bit is.
I also planted a hedge, which keeps a lot of them at bay.
I've since made a border along one edge [ I had an extension built ] so all of that was originally grass, and the hedge was also part of the grass. There's another border which has shrubs.
I've also got daffs and primulas etc, and a little area right down at the front where I'm doing wildflowers. I just started that last year.
If you're going to have bulbs, the best way is to just dig out a hole in the grass and put the bulb in, or peel back a small section of turf if it's for a little group of bulbs. As long as you haven't used any products fro about 6 weeks or so, it'll be fine.
If you want a proper bed or border, it's better to dig up a whole section of turf to plant into. That's what I did for all my planting. The turf can be stacked upside down for later use in borders. Mine has all been re used