Multiple choice

The product of real roots of the equation $|x|^{\tfrac {6}{5}} - 26|x|^{\tfrac {3}{5}} - 27 = 0$ is

  1. $-3^{10}$
  2. $-3^{12}$
  3. $-3^{\frac {12}{5}}$
  4. $-3^{\frac {21}{5}}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let u = |x|^(3/5). The equation is u^2 - 26u - 27 = 0. (u-27)(u+1) = 0. Since |x|^(3/5) must be positive, u = 27. |x|^(3/5) = 27 = 3^3. |x| = (3^3)^(5/3) = 3^5. Thus x = 3^5 or x = -3^5. The product is -3^10.

AI explanation

To solve |x|^(6/5) - 26|x|^(3/5) - 27 = 0, we substitute y = |x|^(3/5), which transforms the equation into y^2 - 26y - 27 = 0. Factoring this quadratic gives (y - 27)(y + 1) = 0, so y = 27 or y = -1, but since y = |x|^(3/5) must be positive, we take y = 27. Reversing the substitution gives |x|^(3/5) = 27, and raising both sides to the power of 5/3 gives |x| = 243. The real roots are 243 and -243, and their product is (243)(-243) = -59049, which matches -3^10.

Keep practicing — related questions