1983 AIME Problems/Problem 3

Problem

What is the product of the real roots of the equation $x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}$ ?

Solution

If we expand this, we get a quartic polynomial, which obviously isn't very helpful.

Instead, we substitute $y$ for $x^2+18x+30$ and our equation becomes $y=2\sqrt{y+15}$ .

Solving for $y$ , we get $y=10$ or $y=-6$ . The second solution gives us non-real roots, so we'll will go with the first. Substituting $x^2+18x+30$ back in for $y$ ,

$x^2+18x+30=10 \Rightarrow x^2+18x+20=0$ . The product of our roots is therefore 20.

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1983 AIME Problems/Problem 3

Problem

Solution

See also