Difference between revisions of "1991 AIME Problems/Problem 7"
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The given finite expansion can then be easily seen to reduce to 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>. | The given finite expansion can then be easily seen to reduce to 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>. | ||
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+ | == Solution 2 == | ||
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+ | <math>x=\sqrt{19}+\underbrace{\frac{91}{\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{x}}}}}}_{x}</math> | ||
+ | |||
+ | |||
+ | <math>x=\sqrt{19}+\frac{91}{x}</math> | ||
+ | |||
+ | <math>x^2=x\sqrt{19}+91</math> | ||
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+ | <math>x^2-x\sqrt{19}-91</math> | ||
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+ | <math>\left.\begin{array}{l}x_1=\frac{\sqrt{19}+\sqrt{383}}{2}\\\\x_2=\frac{\sqrt{19}-\sqrt{383}}{2}\end{array}\right\}A=|x_1|+|x_2|\Rightarrow\sqrt{383}</math> | ||
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+ | <math>\boxed{A^2=383}</math> | ||
== See also == | == See also == |
Revision as of 11:59, 1 August 2017
Contents
Problem
Find , where is the sum of the absolute values of all roots of the following equation:
Solution
Let . Then , from which we realize that . This is because if we expand the entire expression, we will get a fraction of the form on the right hand side, which makes the equation simplify to a quadratic. As this quadratic will have two roots, they must be the same roots as the quadratic .
The given finite expansion can then be easily seen to reduce to the quadratic equation . The solutions are . Therefore, . We conclude that .
Solution 2
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 |
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