answer the above question
Answers
We don't know the two roots. We are starting from a few clues and we should use identities to go on.
Solution: Let the roots α, β. (α≥β to add more clarity)
Given:
A. The difference of roots is 5. ∴α-β=5
B. The difference of their cubes is 215. ∴α³-β³=215
From α³-β³=215 we have (α-β)(α²+αβ+β²)=215.
Divide by α-β and we have α²+αβ+β²=43.
"But what are α+β and αβ?"
Solution: (α-β)²=α²-2αβ+β² will have common parts.
α²-2αβ+β²=25 and α²+αβ+β²=43 have α²+β² common.
Subtract them and we have 3αβ=18. ∴αβ=6
"What is α+β?"
Solution: What should we add to α²+αβ+β² to obtain (α+β)²?
We just add αβ. Now we have (α+β)²=49.
"Should the value become ±7?"
Solution: Use both but find a contradiction.
When their sum is 7. ∴α+β=7
Two solutions are α=6 and β=1. → No contradiction.
When their sum is -7. ∴α+β=-7
Two solutions are α=-1 and β=-6. → No contradiction.