Difference between revisions of "1960 AHSME Problems/Problem 11"

m (See Also)
m (Solution)
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==Solution==
 
==Solution==
If the product of the roots are <math>7</math>, then by Vieta's formulas,  
+
If the product of the roots are <math>7</math>, then by [[Vieta's formulas]],  
 
<cmath>2k^2-1=7</cmath>
 
<cmath>2k^2-1=7</cmath>
 
Solve for <math>k</math> in the resulting equation to get
 
Solve for <math>k</math> in the resulting equation to get
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<cmath>k^2=4</cmath>
 
<cmath>k^2=4</cmath>
 
<cmath>k=\pm 2</cmath>
 
<cmath>k=\pm 2</cmath>
That means the two quadratics are <math>x^2-6x+7=0</math> and <math>x^2+6x+7=0</math>.  Since <math>b^2</math>, <math>a</math>, and <math>c</math> are the same, the discriminant of both is
+
That means the two quadratics are <math>x^2-6x+7=0</math> and <math>x^2+6x+7=0</math>.  Since <math>b^2</math>, <math>a</math>, and <math>c</math> are the same, the [[discriminant]] of both is
 
<math>36-(4 \cdot 1 \cdot 7) = 8</math>.  Because <math>8</math> is not a perfect square, the roots for both are irrational, so the answer is <math>\boxed{\textbf{(D)}}</math>.
 
<math>36-(4 \cdot 1 \cdot 7) = 8</math>.  Because <math>8</math> is not a perfect square, the roots for both are irrational, so the answer is <math>\boxed{\textbf{(D)}}</math>.
  
 
==See Also==
 
==See Also==
 
{{AHSME 40p box|year=1960|num-b=10|num-a=12}}
 
{{AHSME 40p box|year=1960|num-b=10|num-a=12}}

Revision as of 11:44, 14 May 2018

Problem

For a given value of $k$ the product of the roots of $x^2-3kx+2k^2-1=0$ is $7$. The roots may be characterized as:

$\textbf{(A) }\text{integral and positive} \qquad\textbf{(B) }\text{integral and negative} \qquad \\ \textbf{(C) }\text{rational, but not integral} \qquad\textbf{(D) }\text{irrational} \qquad\textbf{(E) } \text{imaginary}$

Solution

If the product of the roots are $7$, then by Vieta's formulas, \[2k^2-1=7\] Solve for $k$ in the resulting equation to get \[2k^2=8\] \[k^2=4\] \[k=\pm 2\] That means the two quadratics are $x^2-6x+7=0$ and $x^2+6x+7=0$. Since $b^2$, $a$, and $c$ are the same, the discriminant of both is $36-(4 \cdot 1 \cdot 7) = 8$. Because $8$ is not a perfect square, the roots for both are irrational, so the answer is $\boxed{\textbf{(D)}}$.

See Also

1960 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
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