Compare the area of the bases. If a base has a greater area, it will take up more space on the shelf

Question

Base area of Box 1: 3x²

Base area of Box 2: 4x² – x

Which box is likely to occupy more space on the shelf? Explain. Try substituting different values for x.

Answer 1

4x^2 -x = 3x^2

x^2 = x

x= 1

4x^2 -x = 4-1 =3

3x^2 = 3

x^2 =1

x = + or -1

same size for x =1

4x^2-x is larger for x>1

x^3 is larger for 0<x<1

x=0, they both disappear

the coefficient of the highest powered term dominates at larger values of x

4(1/2)^2 -1 = 4/4-1= 0, while (1/2)^3 = 1/8 >0

at x=0, both books disappear into a one dimensional stick with zero base

it's a ghost book, the tiniest reed, infinitesmal

for x<0, the x^2-x both books become alive again

at x=-1, they're the same size

-1<x<0 the 4x^2-x book is larger. 4(-1/2)^2 -(-1/2) = 1+1/2 = 3/2 while (-1)^2 = 1<3/2

x<-1 the 4x^2-x book is larger