1964 AMC 12 Problem 30

Attempt Problem 30 of the 1964 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 1964 AMC 12 solutions, or check the answer key.

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30.

$\begin{aligned} (7+4\sqrt3)x^2 &+(2+\sqrt3)x\\ &-2=0, \end{aligned}$

the larger root minus the smaller root is:

$-2+3\sqrt3$

$2-\sqrt3$

$6+3\sqrt3$

$6-3\sqrt3$

$3\sqrt3+2$

Answer: D

Concepts:quadratic radical algebraic manipulation

Difficulty rating: 1670

Small Hint:

Notice that $7+4\sqrt3=(2+\sqrt3)^2$

Big Hint:

For $Ax^2+Bx+C=0,$ the root difference is $\frac{\sqrt{B^2-4AC}}{A}$

Solution:

Let $A=7+4\sqrt3=(2+\sqrt3)^2$ and $B=2+\sqrt3.$ The discriminant is $B^2-4A(-2)=A+8A=9A.$ Hence the root difference is $\frac{\sqrt{9A}}A=\frac3{2+\sqrt3}=6-3\sqrt3.$

Therefore, the correct answer is D.

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