Difference between revisions of "1983 AIME Problems/Problem 3"
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== Problem == | == Problem == | ||
| − | What is the product of the real | + | What is the [[product]] of the [[real]] [[root]]s of the [[equation]] <math>x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}</math>? |
== Solution == | == Solution == | ||
| − | If we expand | + | If we expand by squaring, we get a quartic [[polynomial]], which obviously isn't very helpful. |
Instead, we substitute <math>y</math> for <math>x^2+18x+30</math> and our equation becomes <math>y=2\sqrt{y+15}</math>. | Instead, we substitute <math>y</math> for <math>x^2+18x+30</math> and our equation becomes <math>y=2\sqrt{y+15}</math>. | ||
| − | + | Now we can square; solving for <math>y</math>, we get <math>y=10</math> or <math>y=-6</math>. The second solution gives us non-real roots, so we'll will go with the first. Substituting <math>x^2+18x+30</math> back in for <math>y</math>, | |
<math>x^2+18x+30=10 \Rightarrow x^2+18x+20=0</math>. The product of our roots is therefore 20. | <math>x^2+18x+30=10 \Rightarrow x^2+18x+20=0</math>. The product of our roots is therefore 20. | ||
| − | + | ||
| + | == See also == | ||
* [[1983 AIME Problems/Problem 2|Previous Problem]] | * [[1983 AIME Problems/Problem 2|Previous Problem]] | ||
* [[1983 AIME Problems/Problem 4|Next Problem]] | * [[1983 AIME Problems/Problem 4|Next Problem]] | ||
* [[1983 AIME Problems|Back to Exam]] | * [[1983 AIME Problems|Back to Exam]] | ||
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| − | |||
* [[AIME Problems and Solutions]] | * [[AIME Problems and Solutions]] | ||
* [[American Invitational Mathematics Examination]] | * [[American Invitational Mathematics Examination]] | ||
Revision as of 11:21, 22 January 2007
Problem
What is the product of the real roots of the equation 
?
Solution
If we expand by squaring, we get a quartic polynomial, which obviously isn't very helpful.
Instead, we substitute 
for 
and our equation becomes 
.
Now we can square; solving for , we get 
or 
. The second solution gives us non-real roots, so we'll will go with the first. Substituting back in for ,
. The product of our roots is therefore 20.
