1959 AMC 12 Problem 34

Attempt Problem 34 of the 1959 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 1959 AMC 12 solutions, or check the answer key.

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34.

Let the roots of

x^2-3x+1=0

be

r

and

s.

Then the expression

r^2+s^2

is:

a positive integer

a positive fraction greater than

1

a positive fraction less than

1

an irrational number

an imaginary number

Answer: A

Concepts:Vieta’s Formulas quadratic algebraic manipulation

Difficulty rating: 1110

Small hint:

Use

r+s=3

and

rs=1

Big hint:

Rewrite

r^2+s^2

as

(r+s)^2-2rs

Solution:

Vieta’s formulas give

r+s=3

and

rs=1.

Therefore

\begin{aligned} r^2+s^2 &=(r+s)^2-2rs\\ &=9-2=7, \end{aligned}

which is a positive integer.

Thus, the correct answer is A.