Multiple choice

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

  1. $20$
  2. $24$
  3. $16$
  4. $18$
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A Correct answer
Explanation

Let y = sqrt(x^2 + 18x + 45). Then x^2 + 18x + 30 = y^2 - 15. The equation becomes y^2 - 15 = 2y, or y^2 - 2y - 15 = 0. Factoring gives (y - 5)(y + 3) = 0. Since y must be positive, y = 5. Thus, sqrt(x^2 + 18x + 45) = 5, so x^2 + 18x + 45 = 25, or x^2 + 18x + 20 = 0. The product of the roots is c/a = 20/1 = 20.

AI explanation

Let u equal the quantity x squared plus 18x plus 45, so the equation becomes u minus 15 equals 2 times the square root of u. Squaring both sides results in u squared minus 30u plus 225 equals 4u, which simplifies to u squared minus 34u plus 225 equals 0. Factoring this quadratic gives (u minus 9) times (u minus 25) equals 0, but rejecting u equals 9 due to extraneous solutions leaves u equal to 25. Substituting x squared plus 18x plus 45 equals 25 gives x squared plus 18x plus 20 equals 0, and since the product of the roots is 20, the result is 20.

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