1961 AMC 12 Problem 29

Attempt Problem 29 of the 1961 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 1961 AMC 12 solutions, or check the answer key.

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29.

Let the roots of

ax^2+bx+c=0

be

r

and

s.

The equation with roots

ar+b

and

as+b

is:

x^2-bx-ac=0

x^2-bx+ac=0

x^2+3bx+ca+2b^2=0

x^2+3bx-ca+2b^2=0

\begin{gathered}x^2+bx(2-a)\\{}+a^2c+b^2(a+1)=0\end{gathered}

Answer: B

Concepts:Vieta’s Formulas quadratic algebraic manipulation

Difficulty rating: 1500

Small hint:

Use

r+s=-\frac{b}{a}

and

rs=\frac{c}{a}

Big hint:

Compute the sum and product of

ar+b

and

as+b

Solution:

The transformed roots have sum

a(r+s)+2b=-b+2b=b

and product

\begin{gathered} (ar+b)(as+b)\\ =a^2rs+ab(r+s)+b^2\\ =ac. \end{gathered}

Therefore their monic equation is

x^2-bx+ac=0.

Thus, the correct answer is B.