Shadow of the Colossus/Quadratus - Colossus 2

Reaching Quadratus
The second colossus you are guided towards is Quadratus, the Bull. It is as large and lumbering as the first colossus and can be found by crossing a natural bridge directly north of the Shrine of Worship. You will need to follow a path down towards the valley bottom once over the bridge, and at the bottom approach the sealed off cave. The colossus will reveal itself.

Defeating Quadratus
After the short cut scene, it's time to prep for battle with the beast. Start by riding (if you're still on Agro) a good distance away from the colossus. From there, switch out the hero's sword with his bow, then dismount and slowly approach the beast. It is a good idea to remain cautious with your distance as being step on will lower your health.

The colossus' first weak point is on the bottom of its hooves. There are a couple of strategies that can be used to attack them. The first and probably simplest is to wait for the colossus to rear on its hind legs when about to stomp on the hero, and then fire an arrow at the glowing bottoms. Another technique involves flanking the beast and firing at the back hooves while it walks. Each time it moves its feet, the yellow undersides are briefly revealed. You will need to have proficiency with the bow for aiming and timing. While doable, this technique is by far more difficult.

Regardless of your method, the colossus will fall to one side. Take this opportunity to climb up the hairy upper-regions of its front legs and onto the colossus' back. There are two magic weakpoints, one on Quadratus' head, the other at his rear. Be careful because Quadratus will likely try to shake you off. If this happens, you can always jump onto the furry portion of his back and hold on until he's done. You may also find that your stamina will give out on you whilst traversing the colossus' furry sides. In this case it's best just to go for it and hope you reach where you need to be.

Once you reach a weak point, simply charge the hero's sword and stab it like you did with Valus. It may thrash a bit, but as long as you hold on there should be no problem staying on. After about half his life is drained, the magic weak point will vanish. From this point on, there are very few colossi that only have one magic weak spot (most have two or three).

Since the other weak point will be at the opposite end of Quadratus' body, run along the spinal cord (you may need to do a few jumps to get over the ridges). After a few successive stabs, Quadratus will fall and the hero will be transported back to the shrine for his next assignment.

Hard Mode Strategy
Once the mark on the Quadratus' flank vanishes, hold up the sword. The weakspot will appear on one of furry sides used to climb up it. If the points gather at the left, head towards his left side (and vice versa). Now climb down until you see his weakspot and finish the beast off once and for all. If you fall off from fatigue (or another reason), shoot one of his hooves again and climb back up.

Time Attack Strategy
Quadratus is more difficult than he seems at first, especially in hard mode where you have precious little stamina. Don't bother with Agro here, just run to your left and get directly in front of Quadratus and quickly hit him with an arrow in the face. He should stop and rear at you. Hit his left (your right) hoof with an arrow and he'll fall on his right side. Quickly grab on to his fur and make your way up his left leg. You must use the shelf here on his shoulder to rest, don't try to head back yet because you won't have the strength to make it. Kneel down to gather your strength faster. Ready your sword then leap across his body and try to grab on about half way back. If you've done it correctly, you should end up close to the first vital on his side. Waste no time hitting this vital, you need to finish it off before you have too little stamina to climb back toward his rear. Fortunately Quadratus won't do much to stop you.

Once you've defeated that vital, hopefully you have enough energy to make it on to his backside. If you fall off, it'll be nearly impossible to make your time. Stop here and recover your energy on one of the other shelves. Run back and start hitting the rear vital point. Time isn't as critical here because you can climb back up and recover your strength but don't fool around. Once you're done there it should be pretty straightforward. Run up the right side of his back and take the head vital last.