1969 AMC 12 Problem 12

Attempt Problem 12 of the 1969 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1969 AMC 12 solutions, or check the answer key.

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12.

Let

F=\dfrac{6x^2+16x+3m}{6}

be the square of an expression which is linear in

x.

Then

m

has a particular value between:

3

and

4

4

and

5

5

and

6

-4

and

-3

-6

and

-5

Rewrite

F=x^2+\frac83x+\frac m2

Match the constant term to the square of half the linear coefficient

We have

F=x^2+\frac83x+\frac m2.

For this to be a square, it must equal

(x+\frac{4}{3})^2,

whose constant term is

\frac{16}{9}.

Hence

\frac{m}{2}=\frac{16}{9},

m=\frac{32}{9},

which lies between

3

and

4.

Therefore, the correct answer is A.

Problem 12 in other years