If you cast a spell "without paying its mana cost", you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the spell has any mandatory additional costs, you must pay those to cast it.

If the discovered card has in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

When you discover, you must exile cards. The only optional part of the ability is whether you cast the exiled card or put it into your hand.

The mana value of a split card is determined by the combined mana cost of its two halves. If discover allows you to cast a split card, you may cast either half (as long as its mana value is less than or equal to the effect's discover value) but not both halves.

A spell's mana value is determined only by its mana cost. Ignore any alternative costs, additional costs, cost increases, or cost reductions.

If you can't cast the discovered card (perhaps because there are no legal targets for the spell), you'll put it into your hand.

If you discover an adventurer card, split card, or modal double-faced card, you might be able to cast that card with either set of characteristics depending on the effect's discover value. For example, if you discover 4 and reveal Galvanic Giant (an adventurer card from Wilds of Eldraine with a mana value of 4), you could cast Galvanic Giant, but not Storm Reading (its Adventure, which has a mana value of 7). If you discover 7 and reveal Galvanic Giant, you could cast either Galvanic Giant or Storm Reading.

Geological Appraiser's triggered ability triggers if you cast it from any zone. It doesn't trigger if you put Geological Appraiser onto the battlefield without casting it.

You exile the cards face up. All players will be able to see them.

Some spells and abilities that cause you to discover may require targets. If each target chosen is an illegal target as that spell or ability tries to resolve, it won't resolve and you won't discover.