1955 AHSME Problems/Problem 18

Problem 18

The discriminant of the equation $x^2+2x\sqrt{3}+3=0$ is zero. Hence, its roots are:

$\textbf{(A)}\ \text{real and equal}\qquad\textbf{(B)}\ \text{rational and equal}\qquad\textbf{(C)}\ \text{rational and unequal}\\ \textbf{(D)}\ \text{irrational and unequal}\qquad\textbf{(E)}\ \text{imaginary}$

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 $(x + \sqrt{3})^2=0$ . The distinct solution^ to the equation, as we can clearly see, is $-\sqrt{3}$ , which is $\textbf{(A)} \text{real and equal}$

^The problem pretends to have two solutions.

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1955 AHSME Problems/Problem 18

Problem 18

Solution