If alpha and beta are roots of ay^2 + by + c = 0, Find the value of 1/alpha^4 + 1/beta^4

ay^2 + by + c = 0 has roots α, β. For this equation, we know that

Sum of roots = -b/a
=> α + β = -b/a

Product of Roots = c/a
= αβ = c/a

Using these two results, we'll find out the value of 1/α^4 + 1/β^4

= 1/α^4 + 1/β^4
= (α^4 + β^4) / α^4.β^4
= (α²)² + (β²)² / (αβ)^4
= (α² + β²)² - 2α²β² / (αβ)^4               by using a² + b² = (a + b)² - 2ab in the numerator
= ((α + β)²  - 2αβ)² - 2(αβ)² / (αβ)^4               by using the same identity again

We will now substitute the values of α + β and αβ which we found earlier
= ((-b/a)² - 2c/a)² - 2(c/a)² / (c/a)⁴
= (b²/a² - 2c/a)² - 2c²/a² / (c/a)⁴
= ((b² - 2ac)/a²)² - 2c²/a² / (c/a)⁴
= (b² - 2ac)²/a⁴ - 2c²/a² / (c/a)⁴
= {((b² - 2ac)² - 2a²c²) / a⁴ } / (c/a)⁴
= (b⁴ - 4ab²c + 4a²c² - 2a²c²) / c⁴
= (b⁴ - 4ab²c + 2a²c²) / c⁴

(b⁴ - 4ab²c + 2a²c²) / c⁴ is our answer.

P.S.- This is a very simple and straightforward question. But you need to be attentive and careful while carrying out the steps because one slight mistake can spoil all your work and you don't have any direct way to cross check the solution either.