Difference between revisions of "1959 AHSME Problems/Problem 34"
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Let the roots of <math>x^2-3x+1=0</math> be <math>r</math> and <math>s</math>. Then the expression <math>r^2+s^2</math> is: <math>\textbf{(A)}\ \text{a positive integer} \qquad\textbf{(B)}\ \text{a positive fraction greater than 1}\qquad\textbf{(C)}\ \text{a positive fraction less than 1}\qquad\textbf{(D)}\ \text{an irrational number}\qquad\textbf{(E)}\ \text{an imaginary number}</math> | Let the roots of <math>x^2-3x+1=0</math> be <math>r</math> and <math>s</math>. Then the expression <math>r^2+s^2</math> is: <math>\textbf{(A)}\ \text{a positive integer} \qquad\textbf{(B)}\ \text{a positive fraction greater than 1}\qquad\textbf{(C)}\ \text{a positive fraction less than 1}\qquad\textbf{(D)}\ \text{an irrational number}\qquad\textbf{(E)}\ \text{an imaginary number}</math> | ||
==Solution== | ==Solution== | ||
− | You may recognize that <math>r^2+s^2</math> can be written as <math>(r+s)^2-2rs</math>. Then, by [[Vieta's formulas]], <cmath>r+s=-(-3)=3</cmath>and <cmath>rs=1.</cmath>Therefore, plugging in the values for <math>r+s</math> and <math>rs</math>, <cmath>(r+s)^2-2rs=(3)^2-2(1)=9-2=7.</cmath>Hence, we can say that the expression <math>r^2+s^2</math> is <math>\boxed{\ | + | You may recognize that <math>r^2+s^2</math> can be written as <math>(r+s)^2-2rs</math>. Then, by [[Vieta's formulas]], <cmath>r+s=-(-3)=3</cmath>and <cmath>rs=1.</cmath>Therefore, plugging in the values for <math>r+s</math> and <math>rs</math>, <cmath>(r+s)^2-2rs=(3)^2-2(1)=9-2=7.</cmath>Hence, we can say that the expression <math>r^2+s^2</math> is <math>\boxed{\textbf{(A)}\ \text{a positive integer.}}</math> |
==See also== | ==See also== | ||
{{AHSME 50p box|year=1959|num-b=33|num-a=35}} | {{AHSME 50p box|year=1959|num-b=33|num-a=35}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 14:43, 21 July 2024
Problem 34
Let the roots of be and . Then the expression is:
Solution
You may recognize that can be written as . Then, by Vieta's formulas, and Therefore, plugging in the values for and , Hence, we can say that the expression is
See also
1959 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 33 |
Followed by Problem 35 | |
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