Below are three sites that offer explanations. However, before that, better is what GotQuestions.org says about how to handle discrepancies.

Question: "Does the Bible contain errors, contradictions, or discrepancies?"

Answer: If we read the Bible at face value, without a preconceived bias for finding errors, we will find it to be a coherent, consistent, and relatively easy-to-understand book. Yes, there are difficult passages. Yes, there are verses that appear to contradict each other. We must remember that the Bible was written by approximately 40 different authors over a period of around 1500 years. Each writer wrote with a different style, from a different perspective, to a different audience, for a different purpose. We should expect some minor differences. However, a difference is not a contradiction. It is only an error if there is absolutely no conceivable way the verses or passages can be reconciled. Even if an answer is not available right now, that does not mean an answer does not exist. Many have found a supposed error in the Bible in relation to history or geography only to find out that the Bible is correct once further archaeological evidence is discovered.

We often receive questions along the lines of “Explain how these verses do not contradict!” or “Look, here is an error in the Bible!” Admittedly, some of the things people bring up are difficult to answer. However, it is our contention that there are viable and intellectually plausible answers to every supposed Bible contradiction and error. There are books and websites available that list “all the errors in the Bible.” Most people simply get their ammunition from these places; they do not find supposed errors on their own. There are also books and websites available that refute every one of these supposed errors. The saddest thing is that most people who attack the Bible are not truly interested in an answer. Many “Bible attackers” are even aware of these answers, but they continue to use the same old shallow attacks again and again.

So, what are we to do when someone approaches us with an alleged Bible error? 1) Prayerfully study the Scriptures and see if there is a simple solution. 2) Do some research using some of the fine Bible commentaries, “Bible defense” books, and biblical research websites. 3) Ask our pastors/church leaders to see if they can find a solution. 4) If there is still no clear answer after steps 1), 2), and 3) are followed, we trust God that His Word is truth and that there is a solution that just simply has not been realized yet (2 Timothy 2:15, 3:16-17).
Recommended Resources: The Big Book of Bible Difficulties by Geisler & Howe

***1. First explanation of Contradictions explained by ANSWERSINGENESIS.ORG

When Jesus entered Capernaum he healed the slave of a centurion. Did the centurion come personally to request Jesus for this? At first glance, there appears to be a contradiction between Matthew 8:5, which seems to imply that the centurion came to Jesus, and Luke 7:3 and 7:6, which say he did not.

The Apparent Contradiction

Now when Jesus had entered Capernaum, a centurion came to Him, pleading with Him, saying, "Lord, my servant is lying at home paralyzed, dreadfully tormented." And Jesus said to him, "I will come and heal him." The centurion answered and said, “Lord, I am not worthy that You should come under my roof. But only speak a word, and my servant will be healed. (Matthew 8:5–8)

So when he [the centurion] heard about Jesus, he sent elders of the Jews to Him, pleading with Him to come and heal his servant. And when they came to Jesus, they begged Him earnestly, saying that the one for whom He should do this was deserving, “for he loves our nation, and has built us a synagogue.” Then Jesus went with them. And when He was already not far from the house, the centurion sent friends to Him, saying to Him, "Lord, do not trouble Yourself, for I am not worthy that You should enter under my roof. (Luke 7:3–6)

The First Possible Solution

In Luke 7:6–8 we see the deployment of the centurion’s friends, who spoke the thoughts of his own heart. Matthew mentions the case, but far more succinctly. Matthew frequently abbreviates his retellings of events. [See Matthew 8:14–15 (cf. Mark 1:29–31); Matthew 9:1–8 (cf. Mark 2:1–12); Matthew 9:18–26 (cf. Mark 5:21–43); and Matthew 11:2–6 (cf. Luke 7:18–23)]. So at first blush, we might conclude from Matthew’s account that the centurion came himself; whereas it is clear there was the intervention of both Jewish elders and friends from Luke’s account. Matthew omits the elders’ praise of the centurion in his narrative, but Matthew does include the centurion’s elders and friends, only he does it indirectly and in a unique way. The clue to it is that old and well-attested maxim of law that what one does by another one does by oneself (often mentioned in Scripture: 2 Samuel 3:18, 14:19; 2 Kings 14:27; Esther 8:8; Ezra 1:7–8). So although the narrative in Matthew does not directly mention the centurion’s friends and elders, he does so indirectly by summing it all up, simply speaking of the friends’/elders’ words as nothing but a projection of the centurion’s pleas. Matthew skips the intermediaries and in verse 10 records Jesus piercing to the thoughts and intents of the centurion’s heart.

The Second Possible Solution

When looking at the two texts in total (Matthew 8:5–13 and Luke 7:1–10), it seems quite likely that when Jesus came near to the centurion's house (Luke 7:6), He was probably within shouting distance. Consequently, when Jesus was talking to the centurion's friends, He either spoke loudly enough that the centurion could hear (and was thus addressing him directly) or he spoke to the friends as people who were direct mouthpieces for the centurion. Luke 7:9 states that while talking to the centurion's friends Jesus marveled at him (i.e., the centurion), so it is quite possible that Jesus was talking to the friends of the centurion, but looking directly at and addressing the thoughts and intentions of the centurion. The centurion may have been mindful that Jesus would be considered ceremonially impure if He came into a Gentile’s house (as mentioned in a different context in Acts 10:28) and therefore stood outside the house, so that if Jesus persisted in coming to the centurion, He would not be defiled. Remember that on another later occasion (Matthew 16:23) Jesus addressed someone directly who was the controlling force behind another, even though they were not present (“get behind me Satan” to Peter).

A Side Note—Differences in Perspective and Audience

In Luke 7:4–6 Jesus listened to and then went with the elders of the Jews, toward the centurion’s house, after hearing their request and their reasons for it without any reluctance; He at once complied, did not hesitate, or raise any objection about it, but went with them freely. Matthew completely omits this detail and discourse and instead focuses on the centurion’s thoughts and motives, not what his friends and benefactors thought of the centurion. Remember that Luke was writing to a Gentile audience. Luke wanted Theophilus to be aware that most Jewish people respected a God-fearing Roman soldier. Matthew writing to a Jewish audience wanted them to know that a Roman centurion was sincerely respectful of Jesus. So when viewed with these different perspectives and audiences in mind, it becomes clear that we should expect different emphases from the writers, but the account itself is non-contradictory.

***2. Did Jesus and the Centurion Speak to Each Other Personally?

by	Jim Estabrook

On one occasion when Jesus entered Capernaum, He was asked to heal a certain centurion’s servant. Skeptics allege that a contradiction exists between Matthew’s account of this story (8:5-13) and Luke’s account (7:1-10). Whereas Matthew’s account says, “a centurion came to Him, pleading with Him” on behalf of his servant, Luke recorded that “he [the centurion—JE] sent elders of the Jews to Him, pleading with Him to come and heal his servant.” Since Matthew seems to indicate that the centurion personally came to talk to Jesus, and Luke’s account says that the centurion sent others to plead with Christ, skeptics contend that the two accounts are in no way harmonious. Rather, they (supposedly) represent an obvious contradiction, and thereby serve as proof that the Bible is not the infallible Word of God.

Those who claim that such differences represent legitimate errors fail to realize that the Bible often gives “credit” to one in authority, even when others do the work. For example, when John wrote, “Pilate took Jesus and scourged Him” (19:1), he simply meant that Pilate ordered it to be done. Likewise, when the text says that Jesus made and baptized more disciples than John, it means that His disciples baptized more than John (John 4:1-2). In fact, the apostle John clarified this when he wrote, “though Jesus Himself did not baptize, but His disciples” (4:2). Throughout the Bible, people are sent to speak on behalf of a person, and sometimes the text indicates that the person in position of authority actually spoke for himself when, in fact, that person was not even present. The liaison that spoke was doing so with his authority. Today, as in times past, courts of law hold that “what a man does through a duly constituted agency, he himself actually and legally does” (Coffman, 1974, p. 105). When the president sends staff members to speak around the world on his behalf, he is the one responsible for the decisions rendered in his absence. In the same way, the centurion sent others to talk to Jesus on behalf of one of his servants. Matthew simply used a common form of speech where one attributes a certain act to a person— an act that is performed not by him, but by his authority (see Boles, 1952, p. 188).

One also must admit that it is possible Matthew and Luke wrote about two different accounts. Although I tend to believe that they were writing about the same incident, it is possible that Jesus had a very similar situation arise in the same town with another centurion, or the same centurion with another servant. Remember, John stated that “there are also many other things that Jesus did, which if they were written one by one, I suppose that even the world itself could not contain the books that would be written” (John 21:25).

Matthew 8:5-13 and Luke 7:1-10 are in no way contradictory. By understanding that Luke simply was more specific than Matthew and that Matthew used a common form of speech (which we still use today), it is clear that the two accounts are harmonious.

***3. Bible Hub - multiple explanations

EXPOSITORY (ENGLISH BIBLE)

Ellicott's Commentary for English Readers

(13) As thou hast believed.—The words were, of course, sent as a message. Better, As thou didst believe—referring to his one great act of faith.

Matthew Henry's Concise Commentary

8:5-13 This centurion was a heathen, a Roman soldier. Though he was a soldier, yet he was a godly man. No man's calling or place will be an excuse for unbelief and sin. See how he states his servant's case. We should concern ourselves for the souls of our children and servants, who are spiritually sick, who feel not spiritual evils, who know not that which is spiritually good; and we should bring them to Christ by faith and prayers. Observe his self-abasement. Humble souls are made more humble by Christ's gracious dealings with them. Observe his great faith. The more diffident we are of ourselves, the stronger will be our confidence in Christ. Herein the centurion owns him to have Divine power, and a full command of all the creatures and powers of nature, as a master over his servants. Such servants we all should be to God; we must go and come, according to the directions of his word and the disposals of his providence. But when the Son of man comes he finds little faith, therefore he finds little fruit. An outward profession may cause us to be called children of the kingdom; but if we rest in that, and have nothing else to show, we shall be cast out. The servant got a cure of his disease, and the master got the approval of his faith. What was said to him, is said to all, Believe, and ye shall receive; only believe. See the power of Christ, and the power of faith. The healing of our souls is at once the effect and evidence of our interest in the blood of Christ.

Barnes' Notes on the Bible

He was healed in that self-same hour - This showed decisively the goodness and power of Jesus. No miracle could be more complete. There could be no imposition or deception.This account, or one similar to this, is found in Luke 7:1-10. There has been a difference of opinion whether the account in Luke refers to the same case as that recorded in Matthew, or whether a second centurion, encouraged by the success of the first, applied to our Saviour in a similar case and manner, and obtained the same success. In support of the supposition that they are different narratives, it is said that they disagree so far that it is impossible to reconcile them, and that it is not improbable that a similar occurrence might take place, and be attended with similar results.
To a plain reader, however, the narratives appear to be the same. They agree in the character of the person, the place, and apparently the time; in the same substantial structure of the account; in the expression of similar feelings, the same answers, and the same result. It is very difficult to believe that all these circumstances would coincide in two different stories.
They differ, however. Matthew says that the centurion "came himself." Luke says that he at first sent elders of the Jews, and then his particular friends. He also adds that he was friendly to the Jews, and had built them a synagogue. An infidel will ask whether there is not here a palpable contradiction. In explanation of this, let it be remarked:
1. That the fact that the centurion came himself, supposing that to have been the fact, is no evidence that others did not come also. It was "in" the city. The centurion was a great favorite, and had conferred on the Jews many favors, and they would be anxious that the favor which he desired of Jesus should be granted. At his suggestion, or of their own accord, his Jewish friends might apply to Jesus, and press the subject upon him, and be anxious to represent the case as favorably as possible. All this was probably done, as it would be in any other city, in considerable haste and apparent confusion; and one observer might fix his attention strongly on one circumstance, and another on another. It is not at all improbable that the same representation and request might have been made both by the centurion and his friends. Matthew might have fixed his eye very strongly on the fact that the centurion came himself, and been particularly struck with his deportment; and Luke on the remarkable zeal shown by the friends of a pagan, the interest they took in his welfare, and the circumstance that he had done much for them. Full of these interesting circumstances, he might comparatively have overlooked the centurion himself. But,
2. It was a maxim among the Jews, as it is now in law, "that what a man does by another, he does himself." So, in Mark 10:35, James and John are represented as coming to the Saviour with a request: in Matthew 20:20, it appears that they presented their request through their mother. In John 4:1, Jesus is said to baptize, when, in fact, he did not do it himself, but by his disciples. In John 19:1, Pilate is said to have scourged Jesus; but he certainly did not do it with his own hands. In the case of the centurion, Matthew narrates what occurred very briefly; Luke goes more into detail, and states more of the circumstances. Matthew was intent on the great leading facts of the cure. He was studious of brevity. He did not choose to explain the particular circumstances. He says that the centurion "made the application" and received the answer. He does not say whether by himself or by "an agent." Luke explains particularly "how" it was done. There is no more contradiction, therefore, than there would be if it should be said of a man in a court of law that he came and made application for a new trial, when the application was really made by his lawyer. Two men, narrating the fact, might exhibit the same variety that Matthew and Luke have done, and both be true. It should never be forgotten that "the sacred narrative of an event is what it is stated to be by all the sacred writers; as the testimony in a court in which a case is decided is what is stated by all the credible witnesses, though one may have stated one circumstance and another another."
One thing is most clearly shown by this narrative: that this account was not invented by the evangelists for the sake of imposition. If it had been, they would have "agreed in all the circumstances."

Jamieson-Fausset-Brown Bible Commentary

Mt 8:5-13. Healing of the Centurion's Servant. ( = Lu 7:1-10).This incident belongs to a later stage. For the exposition, see on [1234]Lu 7:1-10.

Matthew Poole's Commentary

Luke saith, And they that were sent, returning to the
house, found the servant whole that had been sick, Luke 7:10; so as it seemeth that what Christ said unto the centurion, must be interpreted, to those whom the centurion sent in his name. Go your way, your business is done, the centurion’s faith hath obtained, it shall be done to him as he hath believed; where believing must signify, a certain persuasion of the power of Christ, with a relying on this mercy and goodness. The proximate object of faith is some word of God. How far this centurion was acquainted with the oracles of the Old Testament (though he lived amongst the Jews, and, as appears by his building a synagogue for them, Luke 7:5, had a kindness for their religion) I cannot tell. It is most probable that he had some immediate revelation of God concerning Christ, which he is here said to have believed, and to have had a full persuasion of and trusted in. All revelations of God are the object of faith, though the Scriptures, being now written, are to us that have them the tests and touchstones to try such impressions by.
As thou believest, not because thou believest. Our faith is not meritorious of the least mercies, built is an exercise of grace which gives glory to God, and receiveth the reward not of debt but of grace. The miracle appeared in that the disease was of an incurable nature, and the cure was wrought without application of means, and in such a moment of time as means, though used, could not have wrought it.

Gill's Exposition of the Entire Bible

And Jesus said unto the centurion,.... Christ having finished the digression, returns an answer to the centurion, agreeably to his desire, saying to him,go thy way; not as displeased with him, but as granting his request: for it follows,
and as thou hast believed, so be it done unto thee. As he had faith to believe, that Christ could cure his servant by a word speaking, it was done accordingly. Christ by his almighty "fiat" said, let him be healed, and he was healed: just as God in the creation said, "let there be light, and there was light". He does not say according to thy prayer, or according to thy righteousness, and goodness, but according to thy faith: and it is further to be observed, that this cure was wrought, not so much for the sake of the servant, as his master; and therefore Christ says, "be it done unto thee"; let him be healed for thy sake, and restored unto thee, to thy use, profit, and advantage.
And his servant was healed in the selfsame hour, at the very exact time, even in that moment. Some copies add, "and when the centurion returned to his house, in the selfsame hour he found his servant healed"; which the Ethiopic version has, and it agrees with Luke 7:10.

Geneva Study Bible

And Jesus said unto the centurion, Go thy way; and as thou hast believed, so be it done unto thee. And his servant was healed in the selfsame hour.

EXEGETICAL (ORIGINAL LANGUAGES)

Meyer's NT Commentary

Matthew 8:13. Ἐν τῇ ὥρᾳ ἐκ.] ὥρᾳ is emphatic. In the very hour in which Jesus was uttering these words, the slave became whole, and that through the divine power of Jesus operating upon him from a distance, as in John 4:46 ff. The narrative is to be explained neither by a desire to present an enlarging view of the miraculous power of Jesus (Strauss), nor as a parable (Weisse), nor as a historical picture of the way in which God’s word acts at a distance upon the Gentiles (Volkmar), nor as being the story of the woman of Canaan metamorphosed (Bruno Bauer); nor are we to construe the proceeding as the providential fulfilment of a general but sure promise given by Jesus (Ammon), or, in that case, to have recourse to the supposition that the healing was effected through sending an intermediate agent(Paulus). But if, as is alleged, Jesus in His reply only used an affirmation which was halfwaybetween a benediction depending on God and the faith of the house, and a positive act (Keim), it is impossible to reconcile with such vagueness of meaning the simple imperative and the no less impartial statement of the result. Moreover, there exists as little a psychical contact between the sick man and Jesus, as at the healing of the daughter of the woman of Canaan, Matthew 15:22, but the slave was cured in consideration of the centurion’s faith.

Expositor's Greek Testament

Matthew 8:13. ὕπαγε, etc.: compressed impassioned utterance, spoken under emotion = Go, as thou hast believed be it to thee; cure as thorough as thy faith. The καὶ before ὡς in T. R. is the addition of prosaic scribes. Men speaking under emotion discard expletives.
Weizsäcker (Untersuchungen über die Evang. Gesch., p. 50) remarks on the felicitous juxtaposition of these two narratives relatively to one another and to the Sermon on Mount. “In the first Jesus has to do with a Jew, and demands of him observance of the law. In this respect the second serves as a companion piece, the subject of healing being a heathen, giving occasion for a word as to the position of heathens. The two combined are happily appended to a discourse in which Jesus states His attitude to the law, forming as complements of each other a commentary on the statement.”

Bengel's Gnomen

Matthew 8:13. Ὡς ἐπίστευσας, as thou hast believed) A bountiful concession.

Pulpit Commentary

Verse 13. - Matthew only. The parallel passage, Luke 7:10, gives the result found by the messengers on their return. And Jesus said unto the centurion, Go thy way; and (omitted by the Revised Version) as thou hast believed, so be it done unto thee. As. Not strictly proportionate, but in the same way as (Matthew 6:12; Matthew 18:33) thou hast now believed, be it done unto thee. And his servant was healed in the selfsame (Revised Version, in that) hour.

Vincent's Word Studies

Was healed (ἰάθη)Note that the stronger word of the centurion (Matthew 8:8) is used here. Where Christ tends, he heals.

Bible Study

Sunday, April 29, 2018

Daniel

Sunday, January 28, 2018

Jude

Friday, January 5, 2018

Is there a discrepancy between the Luke and Matthew centurion stories

The Apparent Contradiction

The First Possible Solution

The Second Possible Solution

A Side Note—Differences in Perspective and Audience

***2. Did Jesus and the Centurion Speak to Each Other Personally?