Difference between revisions of "2013 AMC 10B Problems/Problem 19"
Inferno137 (talk | contribs) (→Problem) |
Inferno137 (talk | contribs) (→Problem) |
||
Line 3: | Line 3: | ||
<math> \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} </math> | <math> \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} </math> | ||
+ | |||
+ | |||
I didn't really understand this problem... | 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 as (b-x) and c as (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 -b +/- \sqrt{b^2-4ac}\/2a. 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, or -1/2-\sqrt{3}\. Multiplying by the conjugate gives -2-\sqrt{3}\. | If there is one root, the discriminant is 0. Therefore, b^2-4ac=0. Let's define a as (b-x) and c as (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 -b +/- \sqrt{b^2-4ac}\/2a. 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, or -1/2-\sqrt{3}\. Multiplying by the conjugate gives -2-\sqrt{3}\. |
Revision as of 21:35, 25 February 2013
Problem
The real numbers form an arithmetic sequence with
The quadratic
has exactly one root. What is this root?
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 as (b-x) and c as (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 -b +/- \sqrt{b^2-4ac}\/2a. 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, or -1/2-\sqrt{3}\. Multiplying by the conjugate gives -2-\sqrt{3}\.