"It does not make sense to buy good/full range speakers for home theater and then to send all the low frequency to the sub."
When one uses full-range speakers with good bass response, one simply doesn't 'send all the low frequency to the sub', one leaves the low frequencies that are indeed present in the other 5 channels in those channels. The result is more directionality to the overall sound AND a less-loaded LFE woofer.
"I believe a good sub with cheap speakers will do the job as well."
Small speakers and 1 sub probably will never sound as good as 5 full-range speakers and a subwoofer.
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When one uses full-range speakers with good bass response, one simply doesn't 'send all the low frequency to the sub', one leaves the low frequencies that are indeed present in the other 5 channels in those channels. The result is more directionality to the overall sound AND a less-loaded LFE woofer.
"I believe a good sub with cheap speakers will do the job as well."
Small speakers and 1 sub probably will never sound as good as 5 full-range speakers and a subwoofer.
.