2013 AMC 10B Problems/Problem 19

Revision as of 20:42, 25 February 2013 by Inferno137 (talk | contribs) (Problem)

Problem

The real numbers $c,b,a$ form an arithmetic sequence with $a\ge b\ge c\ge 0$ The quadratic $ax^2+bx+c$ has exactly one root. What is this root?

$\textbf{(A)}\ -7-4\sqrt{3}\qquad\textbf{(B)}\ -2-\sqrt{3}\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ -2+\sqrt{3}\qquad\textbf{(E)}\ -7+4\sqrt{3}$


I didn't really understand this problem...

If there is one root, the discriminant is 0. Therefore, $b^2-4ac=0$. Let's define $a=(b-x)$ and $c=(b+x)$. If $b^2-4ac=0$, by substitution, $b^2-4(b-x)(b+x)=0$ and distributing, $b^2-4b^2+4x^2=0$. Thus, through simple arithmetic, $4x^2=3b^2$ and $2x=\sqrt{3}\/b$. Then, the root, by quadratic formula, is $\frac{-b+\sqrt{b^2-4ac}}\{2a}.$ (Error compiling LaTeX. Unknown error_msg) Since the discriminant is assumed as zero, the root will be $-b/2a$, or $-b/2(b-x)$ or $-b/2(b-\sqrt{3}\b/2)$. This is equal to $-b/2b-\sqrt{3}\b$ (Error compiling LaTeX. Unknown error_msg) or $-1/2-\sqrt{3}$. Multiplying by the conjugate gives $-2-\sqrt{3}$.