Systematic Theology is a discipline of theological study, that is the study God.
A new Christian naturally hungers to be taught, like a new baby hungers and thirsts for milk (I Pet. 2:2). Too many Christians outgrow this and think they know enough. All Christians should know Bible doctrine, not just the preachers and theologians. Ignorance is no virtue. God often says, “I do not want you to be ignorant” (Rom. 1:13, 11:25; I Cor. 1:8; I Thess. 4:14. Cf. 2 Pet. 3:8). Willful ignorance is sin (2 Pet. 3:5). There are no acceptable excuses, such as “I’m too busy, I’m not a great intellectual, It’s not that important, etc.” Every Christian can and should have a strategic grasp of the Bible and basic Christian doctrine.
The following are quotations from well known theologians regarding the purpose of systematics:
One of the disciplines Systematic theology is any study that answers the question, "What does the whole Bible teach us today?" about any given topic.
Wayne Grudem (Systematic Theology, pg. 13)
We need theology for the sake of people. Theology is the application of the Word by persons to the world and all areas of human life.
John Frame (Salvation Belongs to the Lord, pg. 72)
Theology...the church's reflection on God's performative action in word and deed and its own participation in the drama of redemption.
Michael Horton (Covenant and Eschatology, pg. 4)
Theology is the intellectual, relational and spiritual act of knowing God through the disciplined and communal practice of understanding and applying His word to all areas of life.
Every Christian should be a theologian.
Basic Christian Doctrine
Although the following is not an example of systematic theology per se, it is nevertheless a good example of how a system may develop after gathering teaching from the entire Holy Bible. This summary has been compiled by Mr Matt Slick cite 
1. There is only one God - Isaiah 43:10; 44:6,8; John 17:3; 1 Cor. 8:5-6; Gal. 4:8-9
2. God is a Trinity - 2 Cor. 13:14; 1 Pet. 1:2
3. There are no Gods before or after God - Isaiah 43:10
4. God knows all things - 1 John 3:20
5. God is all powerful - Psalm 115:3
6. God is everywhere - Jer. 23:23,24
7. God is sovereign - Zech. 9:14; 1 Tim. 6:15-16
8. God is spirit - John 4:24
9. God created all that exists - Gen. 1:1; Isaiah 44:24
10. Spirit does not have a body of flesh and bones - Luke 24:39
11. God has always been God - Psalm 90:2
12. Jesus is God - John 1:1,14; 10:30-33; 20:28; Col. 2:9; Phil. 2:5-8; Heb. 1:8
13. Jesus became a man - Phil. 2:5-8
14. Jesus has two natures: divine and human - Col. 2:9; 1 Tim. 2:5
15. Jesus was sinless - 1 Pet. 2:22
16. Jesus is the only way to God the Father - John 14:6; Matt. 11:27; Luke 10:22
17. The Holy Spirit is God - Acts 5:3-4
18. The Holy Spirit is not a force. He is alive - Acts 13:2
19. The Bible is inspired by God - 2 Tim. 3:16
20. All people have sinned - Rom. 3:23, 5:12
21. Man did not evolve, he was created - Gen. 1:26
22. Adam and Eve were real people - Gen. 3:20; 5:1; 1 Tim. 2:13
23. Death entered the world because of Adam's sin - Rom. 5:12-15
24. Sin separates us from God - Isaiah 59:2
25. Jesus died for all our sins - 1 John 2:2; 2 Cor. 5:14; 1 Pet. 2:24
26. Jesus' sacrifice was a substitution, for us - 1 Pet. 2:24
27. Jesus rose from the dead in His physical body - John 2:19-21
28. Those who reject Jesus will go to Hell - Rev. 20:11-15
29. Hell is a place of fiery punishment - Matt. 25:41; Rev. 19:20
30. Hell is eternal - Matt. 25:46
31. The unsaved go to hell forever - Rev. 21:8
32. Salvation is a free gift of God - Rom. 4:5; 6:23; Eph. 2:8-9
33. The Bible is the Word of God - 2 Tim. 3:16
34. Jesus will return visibly to earth - Acts 1:11
35. Christians will be raised from the dead when Jesus returns - 1 Thess. 4:14-17
36. There will be a rapture (being caught up into the clouds with Jesus) - 1 Thess. 4:14-17
37. There will be a final judgment - 2 Pet. 3:7
38. The damned will be thrown into a lake of fire - Rev. 20:15
39. Satan will be cast into the lake of fire - Rev. 20:10
40. There will be a new heavens and a new earth - 2 Pet. 3:13; Rev. 21:1