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Difference between revisions of "2010 AIME II Problems/Problem 7"
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== Solution ==
 
== Solution ==
 
set <math>w=x+yi</math>, so <math>x_1 = x+(y+3)i</math>, <math>x_2 = x+(y+9)i</math>, <math>x_3 = 2x-4+2yi</math>.
 
set <math>w=x+yi</math>, so <math>x_1 = x+(y+3)i</math>, <math>x_2 = x+(y+9)i</math>, <math>x_3 = 2x-4+2yi</math>.
 +
 
Since <math>a,b,c\in{R}</math>, the imaginary part of a,b,c must be 0.
 
Since <math>a,b,c\in{R}</math>, the imaginary part of a,b,c must be 0.
 +
 
Start with a, since it's the easiest one to do:
 
Start with a, since it's the easiest one to do:
 +
 
<math>y+3+y+9+2y=0, y=-3</math>
 
<math>y+3+y+9+2y=0, y=-3</math>
 +
 
and therefore:
 
and therefore:
 +
 
<math>x_1 = x</math>, <math>x_2 = x+6i</math>, <math>x_3 = 2x-4-6i</math>
 
<math>x_1 = x</math>, <math>x_2 = x+6i</math>, <math>x_3 = 2x-4-6i</math>
 +
 
now, do the part where the imaginery part of c is 0, since it's the second easiest one to do:  
 
now, do the part where the imaginery part of c is 0, since it's the second easiest one to do:  
 
<math>x(x+6i)(2x-4-6i)</math>, the imaginery part is: <math>6x^2-24x</math>, which is 0, and therefore x=4, since x=0 don't work,
 
<math>x(x+6i)(2x-4-6i)</math>, the imaginery part is: <math>6x^2-24x</math>, which is 0, and therefore x=4, since x=0 don't work,
 +
 
so now, <math>x_1 = 4, x_2 = 4+6i, x_3 = 4-6i</math>
 
so now, <math>x_1 = 4, x_2 = 4+6i, x_3 = 4-6i</math>
 +
 
and therefore:
 
and therefore:
 +
 
<math>a=-12, b=84, c=-208</math>, and finally, we have <math>|a+b+c|=|-12+84-208|=136</math>.
 
<math>a=-12, b=84, c=-208</math>, and finally, we have <math>|a+b+c|=|-12+84-208|=136</math>.

Revision as of 12:06, 3 April 2010

Problem 7

Let $P(z)=x^3+ax^2+bx+c$, where a, b, and c are real. There exists a complex number $w$ such that the three roots of $P(z)$ are $w+3i$, $w+9i$, and $2w-4$, where $i^2=-1$. Find $|a+b+c|$.

Solution

set $w=x+yi$, so $x_1 = x+(y+3)i$, $x_2 = x+(y+9)i$, $x_3 = 2x-4+2yi$.

Since $a,b,c\in{R}$, the imaginary part of a,b,c must be 0.

Start with a, since it's the easiest one to do:

$y+3+y+9+2y=0, y=-3$

and therefore:

$x_1 = x$, $x_2 = x+6i$, $x_3 = 2x-4-6i$

now, do the part where the imaginery part of c is 0, since it's the second easiest one to do: $x(x+6i)(2x-4-6i)$, the imaginery part is: $6x^2-24x$, which is 0, and therefore x=4, since x=0 don't work,

so now, $x_1 = 4, x_2 = 4+6i, x_3 = 4-6i$

and therefore:

$a=-12, b=84, c=-208$, and finally, we have $|a+b+c|=|-12+84-208|=136$.

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