1992 AHSME Problems/Problem 28
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 | ||
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