2013 AMC 10B Problem 19

Question

The real numbers

c,

b,

a

form an arithmetic sequence with

a \geq b \geq c \geq 0.

The quadratic

ax^2+bx+c

has exactly one root. What is this root?

-7-4\sqrt{3}

-2-\sqrt{3}

-1

-2+\sqrt{3}

-7+4\sqrt{3}

Write the arithmetic sequence as

b-d,b,b+d

Use the zero discriminant condition

Let the common difference be

d

, so

c=b-d

and

a=b+d

.

A quadratic with exactly one real root has discriminant

0

, so

b^2-4ac=0

. Substituting gives

b^2=4(b+d)(b-d)=4b^2-4d^2

, hence

4d^2=3b^2

.

Since

b,d\ge0

,

2d=\sqrt3 b

. The double root is

\frac{-b}{2a}=\frac{-b}{2(b+d)}=\frac{-1}{2+\sqrt3}=-2+\sqrt3

.

Thus, the correct answer is D .

Problem 19 in other years