3U equation involving roots

[Jefferson]

Hi, could I please receive some help for the following questions (in attachment).

If the roots of the equation x³ + px² + qx + r = 0 are consecutive terms of a geometric series, prove that q³ = p³r.
Show that this condition is satisfied for the equation 8x³ - 100x² + 250x - 125 = 0 and solve this equation.

I've also added my solution.
Is there a better approach that I could've taken at any steps? Please show me how to set this question out.
Also the final part when getting the answer, ±, is kinda confusing. Please clarify.

Thanks!

[fun_jirachi]

Hey there!

Your solution is great! I think a better way to do this is to use the roots a/b, a and ab to assist in computations for product of roots (so you just get a^3) but it's otherwise okay! There's not much different I would've done, except for the method I'd use (very similar, but the working out lines are slightly different). When you get to the last part with the ±, note that 2+sqrt3 is equal to the inverse of 2-sqrt3 ie (2-sqrt3)^-1 (you'll see why when you do the working out, that's why it works). When you do the a/b, a and ab method it disappears and makes a lot more sense quicker, but essentially it doesn't really matter which one we pick. It's a cubic so it's limited to three roots, and you already have one, and the other two are some value divided into the first, and the first multiplied by that same value.

Not the best explanation I could give, but hope this helps anyway!

[Jefferson]

Hey there!

Your solution is great! I think a better way to do this is to use the roots a/b, a and ab to assist in computations for product of roots (so you just get a^3) but it's otherwise okay! There's not much different I would've done, except for the method I'd use (very similar, but the working out lines are slightly different). When you get to the last part with the ±, note that 2+sqrt3 is equal to the inverse of 2-sqrt3 ie (2-sqrt3)^-1 (you'll see why when you do the working out, that's why it works). When you do the a/b, a and ab method it disappears and makes a lot more sense quicker, but essentially it doesn't really matter which one we pick. It's a cubic so it's limited to three roots, and you already have one, and the other two are some value divided into the first, and the first multiplied by that same value.

Not the best explanation I could give, but hope this helps anyway!

Your explanation makes perfect sense.
Thank you Jirachi.