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

Answer: E

Concepts:simultaneous equations cube roots of unity complex solutions

Difficulty rating: 1720

Small Hint:

Rewrite the second equation as $y(x+1)=1$

Big Hint:

Substitute $y=(x+1)^2$ to obtain a cubic in $x+1$

Solution:

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.

1955 AMC 12 Problem 43

Answer: E

Difficulty rating: 1720

Small Hint:

Big Hint:

Solution:

Problem 43 in Other Years