Hi all, here is a little guide with the intention to make everyone happy :D
leave a comment if you like, and if you dont, well go away :P

Pros / Cons

Pros:
+Great gap-closer/escape mechanism through Leap Strike
+Low cooldowns late game
+Can be tanky with little armor/magic resist
+ Grandmaster's Might is a large boost to your defenses
+Strong Duelist once you get ultimate
+Strong anti-carry .

. Cons:
-High Cooldowns early game
-Weaker early game before getting Grandmaster's Might
-Only CC is Counter Strike
-Not very tanky without Grandmaster's Might active
-No built in sustain

Runes

Runes can significantly change if you fail or not. Possible choices are written below, followed with few setups.

Runes are sorted from most favourites to those less ones.

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RECOMMENDED RUNE PAGE


+ 8,55 AD (easier last hitting, bonus damage for Leap Strike)
+ 12,69 armor (early game defense)
+ 24,3 MR at lvl 18 (late game defense)
+ 78 HP (early game boost, if you go jungle, take Greater Quintessence of Vampirism)
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MARKS



Recommended: Greater Mark of Strength - flat AD bonus, 9 x 0,95 = 8,55 bonus AD

Viable: Greater Mark of Desolation - 14,94 point of armor penetration plus 6 points from Sunder plus 10% from Weapon Expertise will make your autoattacks penetrate most armor of squishes or at least reduce some armor from more tough targets.

Viable: Greater Mark of Destruction - with whole set of this runes, you will have 14,13 combined penetration instead of 8,55 magic penetration OR 14,94 armor penetration.

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SEALS




Recommended: Greater Seal of Resilience - 9 x 1,41 = 12,69 armor

Viable: Greater Seal of Fortitude - 9 x 5,35 HP = 48,15 HP right from the start of the game

Viable: Greater Seal of Vitality - 9 x 19,5 HP = 175 HP at lvl 18, lvl bonus is 9,72 HP plus 6 HP from Durability makes cca 16 additional HP per lvl

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GLYPHS




Recommended: Greater Glyph of Shielding - stacking MR runes for late game MR, Jax gets very low lvl bonus for MR

Viable: Greater Glyph of Celerity - 9 x 0,05% CDR per lvl = 8,62% on lvl 18

Viable: Greater Glyph of Potency - 9 x 0,99 AP = total 8,91 AP

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QUINTESSENCES



Recommended: Greater Quintessence of Fortitude + 78 HPs

Recommended: Greater Quintessence of Vampirism +6% lifesteal

Viable: Greater Quintessence of Potency 3 x 4,95 AP = 14,85 AP

Viable: Greater Quintessence of Desolation 3 x 3,33 ArPen = 10 ArPen (plus 15 points from 9x
Greater Mark of Desolation plus 6 points from Sunder plus 10% from Weapon Expertise will make your autoattacks penetrate even more armor of squishes or reduce some armor from more tough targets.

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Few notes:
Since passive no longer provides HP from bonus AD and AP, the runes dont have to maximize the starting potential. Now they can focus two things: 1) boost early game (i.e. HP runes) and 2) boost late game (lifesteal). Before the patch was great to take AD/AP runes and benefit from bonus HP, but now the situation is quite different.

Summoner Spells

Just like every character, there are some summoner spells that help Jax and some that are completely and utterly useless on him as well. Then there are those other spells that are atleast reasonable in some way. I will now list

Most Recommended

Teleport is an important spell for solo top. Early game, you can use it to recall, heal up, and come back with full health and an item advantage over your opponent. It can also help if you want to gank mid or bot lane before going back to top quickly with this spell so that the enemy top laner doesn't have as much time to react and come down. However, it's main purpose is to give your team an advantage when fighting over Dragon or other objectives mid to late game. Both of the top laners are rather isolated in top lane but Teleport allows you to quickly join the fight and give your team an advantage. Also, I'd suggest that you do this compared to protecting your turret since getting Dragon gives 40 more gold per person than a turret does, and you could potentially give your team a champion kill or two while doing this which will add to the gold you get. The main purpose of Teleport is generally to give you a presence in teamfights even when you were just farming top lane.

Ghost is also a good spell as it increases your movement speed for an entire 10 seconds, as well as avoid unit collision, allowing you chase, or just finally getting out of range of that Kennen who has been using his Lightning Rush in an attempt to stun you with his Thundering Shuriken. It also helps you chase enemies who have higher health because you will be able to keep up with them for a longer amount of time unlike Flash, and so it is good for when in a situation with Flash, Annie would stun you and skip away, laughing that you couldn't keep up. Instead, you will be able to continue to chase and potentially kill her, or get close to her before she is at her turret and you don't feel safe with her Pyromania stack count on 2 or 3. This will, however, give you more time to attack her turret before she manages to come back.

Exhaust is good mainly when facing a champion like Riven who builds a lot of AD but not much AP under any circumstances, and also against champions who are very mobile. This is mainly because it reduces the AD of a champion by a lot more than the AP. It also decreases the movement speed and attack speed of them so that you are basically certain to deal heavy damage before they have a chance of getting away. However, that doesn't mean that you shouldn't take it against AP, as it will still bring down their AP so that you can win a fight against them, or just to hold them down. It is even more useful when you don't have a jungler with a lot of CC like Shyvana, so that they can keep up with the enemy and ensure that the gank will be successful either by forcing the enemy to burn an escape spell such as Flash or Ghost or resulting in a kill.

There is no exception when you are jungling. You take this or you lose buffs to Nunu. I think you'd rather take Smite, correct? However, never take this when you aren't the jungler. Also, you get 10 bonus gold whenever you smite something because of Summoner's Resolve if you still wanted to take it.

Other viable summoner spells

Flash is a very good multipurpose spell that can be used to get close enough to kill someone on the enemy team because Olaf started running away while a teamfight was happening. A nice way to catch up is to use Flash and then quickly use Leap Strike while you are still within range. Also, it can be used to escape a teamfight that you already know is lost since Annie had found her way back to your fed Graves who was your main source of damage. In this case, Flash away from the main battle and try to run in a different direction than the rest of your team because then you one of two things will happen. The first is that some people keep chasing after the others but one or two come after you. In this case, your team will have a much better chance of having more people still alive and therefore the outcome is better. The second is that nobody comes after you and well, that means that you can come around and help defend the turret in the event that the other team attempts to kill your allies at the turret.

Ignite is mainly used in two circumstances: the first being when you want to finish off an opponent at very low health but can't since they got away, either by being faster or by using Flash or Ghost, and the second when you are facing heavy healers like Swain and Vladimir. Don't underestimate this spell as it still very useful, as it does True Damage however, it is not my personal favorite.

Heal is very useful when you feel that the other team will be poking you a lot in teamfights and when you are going to be harassed a lot in lane which will make it hard for you to farm. This gets rid of that problem and also helps when you are fighting your opponent since you will make them think that you are at low health and can fool them when you use your Heal, but this is mainly something that you should get if you notice that bot lane happens to not take it and instead takes Exhaust and Ignite as their summoner spells, hopefully each of them taking only one of the two.

Surge does provide some decent buffs to your AP and Attack speed, but it isn't really viable in ranked. In normal games, you can use this if you feel like it but definetley not the best choice since it is only meant to help certain champions like Kayle. Do not dare use this in ranked as you will probably end up feeding your opponent and making you lose the game. But if you are looking to have fun, by all means use this in a normal game, especially as Hybrid Jax.

Choices you should never consider

Never even bother with this, this is mainly a spell that your support should get, you should be taking things that are more offensive or defensive, or even both at the same time because YOU ARE NOT A SUPPORT AND JAX SHOULD NEVER BE PLAYED AS SO. This is mainly utility which is what your support should be providing if that person is not already choosing Heal or Exhaust with their Flash.

Clarity is a spell for much more mana hungry champions like Anivia and Singed, but even still, Heal proves to be more useful since you still mainly use only basic attacks when out of mana. However, you normally wouldn't even be out of mana early game unless you were spamming your spells more than neccesary.

Cleanse is also not very useful when you can have a cheap Quicksilver Sash that costs less than 1500 gold and has a much lower cooldown. I just feel that there are much more useful summoner spells than this when most teams don't even have lots of CC like that. If they do, buy Quicksilver Sash if they are focusing you with it.