Difference between revisions of "1958 AHSME Problems/Problem 10"

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==Problem==
 
==Problem==
  
For what real values of <math> k</math>, other than <math> k \equal{} 0</math>, does the equation <math> x^2 \plus{} kx \plus{} k^2 \equal{} 0</math> have real roots?
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For what real values of <math> k</math>, other than <math> k = 0</math>, does the equation <math> x^2 + kx + k^2 = 0</math> have real roots?
  
 
<math> \textbf{(A)}\ {k < 0}\qquad  
 
<math> \textbf{(A)}\ {k < 0}\qquad  

Latest revision as of 03:02, 29 June 2017

Problem

For what real values of $k$, other than $k = 0$, does the equation $x^2 + kx + k^2 = 0$ have real roots?

$\textbf{(A)}\ {k < 0}\qquad  \textbf{(B)}\ {k > 0} \qquad  \textbf{(C)}\ {k \ge 1} \qquad  \textbf{(D)}\ \text{all values of }{k}\qquad  \textbf{(E)}\ \text{no values of }{k}$

Solution

An expression of the form $ax^2+bx+c$ has at least one real root when $b^2-4ac \geq 0$.

Substituting $k$ for $b$ and $k^2$ for $c$, we have

\[k^2-4k^2 \geq 0\]

\[-3k^2 \geq 0\]

but the range of $-3k^2$ is $(-\infty,0]$, so the answer is $\boxed{\text{(E)}}$

See also

1958 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
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All AHSME Problems and Solutions

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