Applied Researchers

Yes, provided Applied Researchers is [Empowered] and the cost belongs to a spell. Its [Empowered] ability grants this discount only while it has the [Empowered] status.^828.1.b.1

[Repeat] is an optional additional cost.^820.1 [Deflect] imposes a mandatory additional cost on a spell or ability that targets the permanent with [Deflect].^809.1.d Discounts can reduce both kinds of additional costs, including to zero.^356.4.f

A [Flow] cost is an alternate cost that replaces a spell's base cost during finalization.^829.1.c.1 Applied Researchers' discount is applied afterward when determining the spell's total cost.^356.1356.4

Because Applied Researchers' discount applies only to spells, it cannot reduce a [Repeat] or [Deflect] cost paid while playing an ability.^{820.1.a809.1.d}

Yes, if Applied Researchers is [Empowered], you can play Shock Blast for 0. Applied Researchers itself satisfies Shock Blast's requirement that you control something that is [Empowered], so both cards' discounts apply.

Both abilities discount Shock Blast's total cost during the Determine Total Cost step of playing it.^356.4 Discounts that apply to the total cost may be applied in any order.^{356.4.d356.4.d.1} A minimum imposed by one discount applies only to that discount.^356.4.e

Cost Sequence

1. Base cost: Shock Blast starts at 3.

2. Applied Researchers reduction: Reduce that cost by 1, leaving 2.

3. Applied Researchers minimum: The reduced cost remains above its minimum of 1.

4. Shock Blast reduction: Reduce the remaining energy cost by 2, leaving 0.

Applying the discounts in the opposite order would leave the spell costing 1, but you may choose the order that results in the lower cost.