Question

answer the above question ​

Answer 1

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.

Therefore, α+β=±7 and αβ=6.

And therefore, the quadratic equation is x²±7x+6=0.