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

Latest revision as of 02:02, 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 $\boxed{\text{(E)}}$

1958 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 1: / Line 1: @@
 ==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?
+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

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

Latest revision as of 02:02, 29 June 2017

Problem

Solution

See also