What is so irreconcilable about them?
Let's examine the two stories point by point.
Matthew doesn't say anything about Zechariah and Elizabeth or the birth of John the Baptist.
He doesn't say anything about the appearance of the angel Gabriel to Mary, or about Mary's visit to Elizabeth and Zechariah.
He doesn't say anything about the taxation in the time of Quirinius (6 AD), or about Joseph and Mary travelling from Nazareth to Bethlehem.
He doesn't say anything about the appearance of the angels to the shepherds.
He doesn't say anything about the proclamation of Jesus by Simeon and Anna in the Temple in Jerusalem. (This story, in Luke 2:22-38, is quite irreconcilable with the story of Herod searching for Jesus to kill him.)
Luke doesn't say anything about Joseph deciding to break off his engagement to Mary, or about an angel telling him in a dream that he must marry her. (Was this before or after Mary's visit to Elizabeth and Zechariah?)
He doesn't say anything about the Magi and the star, or about Herod's inquiries and his instructions to the Magi.
He doesn't say anything about the flight into Egypt or the killing of the children of Bethlehem.
Matthew (2:1) implies that Joseph and Mary were living in Bethlehem when Jesus was born, and he says (Matthew 2:22-23) that they moved into Galilee only to be out of Archelaus's jurisdiction. Luke clearly says (1:26-27, 2:3-4, 2:39-40), that they were living in Nazareth before the birth of Jesus, that they had to go to Bethlehem for the registration, and that they returned to Nazareth after completing the legal formalities.
Matthew was a tax collector for the Romans and a record keeper of the Jews and wrote hid account as a Historian.
Luke was a physician and wrote from a physician's view.
According to modern scholars, all four gospels are anonymous, none of their authors were immediate followers of Jesus, and the gospels of Matthew and Luke were written between 80 and 100 AD, about a hundred years after Jesus's birth. Matthew and Luke both copied from Mark, who says nothing about Jesus's birth being unusual.
They both talk about a virgin conceiving and bringing forth a child.
Jesus himself, as a man, never said anything about his birth being unusual, and there is nothing in the gospels about his mother's opinion on the matter.
Is your problem the fact that they both talked about a virgin giving birth and you don't think that it is possible for a virgin to conceive?
Not necessarily; it is not my subject. The only virgin I am interested in is the one between Leo and Libra. However, according to my understanding of genetics, it is not possible for a woman, with two X chromosomes, to become the mother of a son (who has an X chromosome and a Y chromosome) without the help of a man.
It is, it is possible for any virgin alive today to have a child. Science made that possible.
That is called artificial insemination.
God the Holy Spirit placed God the Son in Mary's womb. The first artificial insemination preformed by God Him Self.
Begging the question. You have to have evidence that there is a God before you can say that he placed Jesus in Mary's womb.
What is it about that, you can't understand?
Enjoy,
Honestly, you Christians amaze me. You are willing to believe something about a woman who lived in a far-away country 2000 years ago that you would not believe of your own sister, daughter or niece.