What is the product of the real roots of the equation x^2+18x+30 =2√x^2+18x+45?

1983 AIME Problem 3

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What is the product of the real roots of the equation

\begin{gathered} x^2+18x+30\\ {}=2\sqrt{x^2+18x+45}? \end{gathered}

Answer

The expression outside the radical is

15

less than the radicand

Set

t=\sqrt{x^2+18x+45}

and solve

t^2-15=2t

Set

t=\sqrt{x^2+18x+45},

t\geq0.

The equation becomes

t^2-15=2t,

(t-5)(t+3)=0.

Thus

t=5,

and

x^2+18x+45=25,

so the real roots satisfy

x^2+18x+20=0.

By Vieta’s formulas, their product is

20.