1967 AMC 12 Problem 35

Question

The roots of

64x^3-144x^2+92x-15=0

are in arithmetic progression. The difference between the largest and smallest roots is:

2

1

\dfrac12

\dfrac38

\dfrac14

Write the roots as

t-d,

t,

and

t+d

Use their sum to find

t,

then use their product to find

d^2

Let the roots be

t-d,

t,

and

t+d.

Their sum is

3t=\frac{144}{64}=\frac{9}{4},

so

t=\frac{3}{4}.

Their product is

t(t^2-d^2)=\frac{15}{64}.

Substituting

t=\frac{3}{4}

gives

d^2=\frac{1}{4},

so the difference between the extreme roots is

2\lvert d\rvert=1.

Therefore, the correct answer is B.

Problem 35 in other years