Multiple choice

The product of the roots of the equation $\displaystyle \sqrt[3]{8+x}+\sqrt[3]{8-x}=1$ is

  1. $-21$
  2. $-189$
  3. $9$
  4. $-5$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let a = cuberoot(8+x) and b = cuberoot(8-x). Then a+b=1 and a^3+b^3 = 16. (a+b)^3 = a^3+b^3+3ab(a+b) => 1 = 16 + 3ab(1) => 3ab = -15 => ab = -5. (ab)^3 = (8+x)(8-x) = 64-x^2. (-5)^3 = -125 = 64-x^2 => x^2 = 189. Roots are sqrt(189) and -sqrt(189). Product = -189.

AI explanation

Let a equal the cube root of (8 plus x) and b equal the cube root of (8 minus x). The given equation becomes a plus b equals 1. Cubing both sides gives a cubed plus b cubed plus 3ab(a plus b) equals 1, and substituting a cubed plus b cubed equals 16 and a plus b equals 1 yields 16 plus 3ab equals 1, so ab equals negative 5. Substituting the original expressions back gives the cube root of ((8 plus x)(8 minus x)) equals negative 5, which simplifies to the cube root of (64 minus x squared) equals negative 5. Cubing both sides results in 64 minus x squared equals negative 125, so x squared equals 189. Therefore, the equation is x squared minus 189 equals 0, and the product of its two roots is negative 189.

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