Difference between revisions of "1991 AIME Problems/Problem 7"
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== Solution == | == Solution == | ||
| − | The given finite expansion can be easily seen that reduces to solve the quadratic equation <math>x_{}^{2}-\sqrt{19}x-91=0</math>. The solutions are <math>x_{\pm}^{}=</math><math>\frac{\sqrt{19}\pm\sqrt{383}}{2}</math>. Therefore, <math>A_{}^{}=\vert x_{+}\vert+\vert x_{-}\vert=\sqrt{383}</math>. | + | The given finite expansion can be easily seen that reduces to solve the quadratic equation <math>x_{}^{2}-\sqrt{19}x-91=0</math>. The solutions are <math>x_{\pm}^{}=</math><math>\frac{\sqrt{19}\pm\sqrt{383}}{2}</math>. Therefore, <math>A_{}^{}=\vert x_{+}\vert+\vert x_{-}\vert=\sqrt{383}</math>. We conclude that <math>A_{}^{2}=383</math>. |
== See also == | == See also == | ||
{{AIME box|year=1991|num-b=6|num-a=8}} | {{AIME box|year=1991|num-b=6|num-a=8}} | ||
Revision as of 21:53, 20 April 2007
Problem
Find 
, where 
is the sum of the absolute values of all roots of the following equation:
Solution
The given finite expansion can be easily seen that reduces to solve the quadratic equation 
. The solutions are 
. Therefore, 
. We conclude that 
.
See also
| 1991 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
