Difference between revisions of "1983 AIME Problems/Problem 3"
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== Problem == | == Problem == | ||
| + | What is the product of the real roots of the equation <math>x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}</math>? | ||
== Solution == | == Solution == | ||
| + | If we expand this, 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>. | ||
| + | |||
| + | 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. | ||
| + | |||
| + | ---- | ||
| + | |||
| + | * [[1983 AIME Problems/Problem 2|Previous Problem]] | ||
| + | * [[1983 AIME Problems/Problem 4|Next Problem]] | ||
| + | * [[1983 AIME Problems|Back to Exam]] | ||
== See also == | == See also == | ||
| − | * [[ | + | * [[AIME Problems and Solutions]] |
| + | * [[American Invitational Mathematics Examination]] | ||
| + | * [[Mathematics competition resources]] | ||
| + | |||
| + | [[Category:Intermediate Algebra Problems]] | ||
Revision as of 22:53, 23 July 2006
Problem
What is the product of the real roots of the equation 
?
Solution
If we expand this, we get a quartic polynomial, which obviously isn't very helpful.
Instead, we substitute 
for 
and our equation becomes 
.
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.
