Difference between revisions of "2013 AMC 10B Problems/Problem 19"
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| − | 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}\qquad\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 20:35, 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}\qquad\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}\.
