1955 AMC 12 Problem 43

Question

The pairs of values of

x

and

y

that are the common solutions of the equations

y=(x+1)^2

and

xy+y=1

are:

3

real pairs

4

real pairs

4

imaginary pairs

2

real and

2

imaginary pairs

1

real and

2

imaginary pairs

Rewrite the second equation as

y(x+1)=1

Substitute

y=(x+1)^2

to obtain a cubic in

x+1

Substitution into

y(x+1)=1

gives

(x+1)^3=1.

The three cube roots of

1

consist of one real root and two nonreal roots. Each determines exactly one corresponding value

y=(x+1)^2.

Therefore there is one real pair and two imaginary pairs.

Thus, the correct answer is E.

Problem 43 in other years