In many verses of the Quran faith and righteous deeds have been mentioned side by side in such a way that shows that they cannot be separated. This is the case because faith and actions perfect each other.

If faith influences the depths of one’s soul its rays will be seen in one’s actions – making them righteous. It is just like a strong light in a room – the light will spread out of the windows. The strong light is faith which shines in one’s heart and reaches one’s eyes, ears, tongue, hands, and feet.
The 11th verse of Sūrah Talāq states: “And those who believe in Allah and work righteousness, He will admit to Gardens beneath which Rivers flow.”[1]
The 55th verse of Sūrah Nūr states: “Allah has promised, to those among you who believe and work righteous deeds, that He will, of a surety, grant them in the land, inheritance (of power).”[2]
Principally, faith is like a root and righteous deeds are fruit. Sweet fruit proves that the roots are healthy and healthy roots cause the growth of useful fruit.
It is possible that faithless people perform righteous deeds from time to time, but it will surely not be forever. The thing that solidifies righteous deeds is faith in the depths of one’s soul which give him the sense of responsibility.