Hi again,
now - the reason why a symmetrical supply is used on such OP circuitries, is the way how this devices work.
The ICs are powered with DC current, but the pure output signal is AC. The chips / ICs, use both DC portentials to operate
with higher efficiency (more power with less losses) and the capability of both (+/-) potentials to build a "neutral point" in the middle.
For this, i've meant to take a look on the suggested supply scheme. "The middle is neutral"

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If you take a look onto an already pre-gained output stage (OPS) you'll maybe see, what i mean.
The power devices are directly connected to both supply rails. The upper devices by the positive, and the lower device by the negative.
(That's how it is shown on most of the standard schematics for OPS).
Now what happens: The applied audio signal will open the junction (by the gate / base current), to enable a defined current
flow through the output devices. The currents now will flow through the upper and the lower device. Here they are connected
- ahh - in the middle. The now amplified audio signal also passes the output devices. Now - here comes the difference between
AC and DC and joins our party.
The DC currents are neutralized in the centre connection, but the amplified audio sinus wave can pass
through the load to the ground connection. That's because the audio signal is an alternating current and not
neutralized by the two of the DC potentials. The DC only transports the AC wave

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The point is:
If you use a single supply (with only + and ground), you'll only be able to get a power efficiency of about 17 - 25%.
The rest are losses!. So - It would never be possible to get 56Watts of audio signal power from a small xt TO220 device

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With a symmetrical / balanced supply you can get a power efficiency of about 50% and little more.
One more very fat advantage is that you don't need signal coupling / decoupling caps, because there is nearly no DC in the
signal path at the points where the AC (signal) goes to the next or OP stage.
Page two of National's LM3875 datasheet shows the basic internals (equivalent scheme).
If you follow the supply rails and the signal path, the things should appear a bit more clear.
Hope that was a little help - have fun

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