Difference between revisions of "1992 AHSME Problems/Problem 28"
(Created page with "== Problem == Let <math>i=\sqrt{-1}</math>. The product of the real parts of the roots of <math>z^2-z=5-5i</math> is <math>\text{(A) } -25\quad \text{(B) } -6\quad \text{(C) } ...") |
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== Solution == | == Solution == | ||
| − | <math>\fbox{B}</math> | + | Applying the quadratic formula gives <cmath>z=\frac{-1\pm\sqrt{21-20i}}{2}</cmath> |
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| + | Let <cmath>\sqrt{21-20i}=a+bi</cmath> where <math>a</math> and <math>b</math> are real. Squaring both sides and equating real and imaginary parts gives <cmath>a^2-b^2=21</cmath> <cmath>2ab=-20</cmath> Substituting <math>b=-\frac{10}{a}</math>, letting <math>a^2=n</math> and solving gives <math>n=25, -4</math>, from which we see that <math>a=5</math> and <math>b=-2</math>. | ||
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| + | Replacing the square root in the quadratic formula with <math>5-2i</math> and simplifying gives the two roots <math>2-i</math> and <math>-3+i</math>. The product of their real parts is <math>-6</math>. The answer is <math>\fbox{B}</math>. | ||
== See also == | == See also == | ||
Latest revision as of 10:14, 1 August 2016
Problem
Let 
. The product of the real parts of the roots of 
is
Solution
Applying the quadratic formula gives ![\[z=\frac{-1\pm\sqrt{21-20i}}{2}\]](http://latex.artofproblemsolving.com/6/4/0/640ed50baa08e19611a5f696c1d39b20d6025677.png)
Let ![\[\sqrt{21-20i}=a+bi\]](http://latex.artofproblemsolving.com/c/9/f/c9f697c18afd4b0692c7350675f7f586b93daa5e.png)
where 
and 
are real. Squaring both sides and equating real and imaginary parts gives ![\[a^2-b^2=21\]](http://latex.artofproblemsolving.com/4/9/6/4969f226b22fadd7f745b5b4a972809570585760.png)
![\[2ab=-20\]](http://latex.artofproblemsolving.com/5/6/a/56a7b0232c236980cc5dce041495e3cc1ef345fd.png)
Substituting 
, letting 
and solving gives 
, from which we see that 
and 
.
Replacing the square root in the quadratic formula with 
and simplifying gives the two roots 
and 
. The product of their real parts is 
. The answer is 
.
See also
| 1992 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 27 |
Followed by Problem 29 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
