Difference between revisions of "1955 AHSME Problems/Problem 18"
Angrybird029 (talk | contribs) (Created page with "== Problem 18== The discriminant of the equation <math>x^2+2x\sqrt{3}+3=0</math> is zero. Hence, its roots are: <math> \textbf{(A)}\ \text{real and equal}\qquad\textbf{(B)}...") |
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<math> \textbf{(A)}\ \text{real and equal}\qquad\textbf{(B)}\ \text{rational and equal}\qquad\textbf{(C)}\ \text{rational and unequal}\\ \textbf{(D)}\ \text{irrational and unequal}\qquad\textbf{(E)}\ \text{imaginary} </math> | <math> \textbf{(A)}\ \text{real and equal}\qquad\textbf{(B)}\ \text{rational and equal}\qquad\textbf{(C)}\ \text{rational and unequal}\\ \textbf{(D)}\ \text{irrational and unequal}\qquad\textbf{(E)}\ \text{imaginary} </math> | ||
==Solution== | ==Solution== | ||
| − | Since the discriminant is zero, there is one distinct root, or, relevant to this question, two equal roots. The fact that only one solution exists means that the equation can be simplified into <math>(x + \sqrt{3})^2=0</math>. The distinct solution | + | Since the discriminant is zero, there is one distinct root, or, relevant to this question, two equal roots. The fact that only one solution exists means that the equation can be simplified into <math>(x + \sqrt{3})^2=0</math>. The distinct solution^ to the equation, as we can clearly see, is <math>-\sqrt{3}</math>, which is <math> \textbf{(A)} \text{real and equal}</math> |
| − | + | ^The problem pretends to have two solutions. | |
| + | == See Also == | ||
| + | {{AHSME box|year=1955|num-b=17|num-a=19}} | ||
| + | |||
| + | {{MAA Notice}} | ||
Latest revision as of 16:55, 2 August 2020
Problem 18
The discriminant of the equation 
is zero. Hence, its roots are:
Solution
Since the discriminant is zero, there is one distinct root, or, relevant to this question, two equal roots. The fact that only one solution exists means that the equation can be simplified into 
. The distinct solution^ to the equation, as we can clearly see, is 
, which is 
^The problem pretends to have two solutions.
See Also
| 1955 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
