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Difference between revisions of "1985 AHSME Problems/Problem 21"
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==Problem==
 
==Problem==
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How many integers <math> x </math> satisfy the equation <math> (x^2-x-1)^{x+2}=1 </math>
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How many integers <math> x </math> satisfy the equation <math> (x^2-x-1)^{x+2}=1 </math>?
  
 
<math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \  } 4 \qquad \mathrm{(D) \  } 5 \qquad \mathrm{(E) \  }\text{none of these} </math>
 
<math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \  } 4 \qquad \mathrm{(D) \  } 5 \qquad \mathrm{(E) \  }\text{none of these} </math>
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<math> x=0, 1 </math>
 
<math> x=0, 1 </math>
  
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However, <math> x=1 </math> gives an odd power of <math> -1 </math>, so this is discarded. Finally, notice that anything to the <math> 0\text{th} </math> power (except for <math> 0 </math>, which is debatable) gives <math> 1 </math>.
+
However, <math> x=1 </math> gives an odd power of <math> -1 </math>, so this is discarded. Finally, notice that anything to the <math> 0\text{th} </math> power (except for <math> 0 </math>, as <math>0^0</math> is undefined) gives <math> 1 </math>.
  
 
<math> x+2=0 </math>
 
<math> x+2=0 </math>

Revision as of 00:04, 3 April 2018

Problem

How many integers $x$ satisfy the equation $(x^2-x-1)^{x+2}=1$?

$\mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ } 5 \qquad \mathrm{(E) \ }\text{none of these}$

Solution

Notice that any power of $1$ is $1$, so $x^2-x-1=1$ would give valid solutions.

$x^2-x-2=0$

$(x-2)(x+1)=0$

$x=2, -1$

Also, $-1$ to an even power also gives $1$, so we check $x^2-x-1=-1$

$x^2-x=0$

$x(x-1)=0$

$x=0, 1$

However, $x=1$ gives an odd power of $-1$, so this is discarded. Finally, notice that anything to the $0\text{th}$ power (except for $0$, as $0^0$ is undefined) gives $1$.

$x+2=0$

$x=-2$

This doesn't make $x^2-x-1=0$, so this is also valid.

Overall, our valid solutions are $x=-2, -1, 0, 2$ for a grand total of $4, \boxed{\text{C}}$.

See Also

1985 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 20 		Followed by
Problem 22
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

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