Find all real solutions of the equation. (enter your answers as a comma-separated list. if there is no real solution, enter no real solution.) x3 − 11x2 + 24x = 0

Hello.  In  x3 − 11x2 + 24x = 0   please use the symbol " ^ " to denote exponentiation:

x^3 − 11x^2 + 24x = 0

First step:  factor out x:    

x^3 − 11x^2 + 24x = 0  =  x(x^2 - 11x + 24) = 0

Second step:  factor the polynomial x^2 - 11x + 24 = 0.

We can see right away that (-3)(-8) = + 24 and that 

-3
-8
---
is equal to -11.  Thus, the factors of   
x^3 − 11x^2 + 24x = 0

are x(x-3)(x-8) = 0.  The solutions are 0, 3 and 8.

B, 0.4 repeats so it is rational and square root of 16 is 4 so that is rational as well.

a. geometricb. g[h] = (3/4)·g[h-1]; g[1] = 384c. g[h] = 384·(3/4)^(h-1)d. g[12] ≈ 16.2183 gallonse. g[12] ≈ 64.8732 quarts

step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt