Multiple choice

Determine the number of distinct integral roots of the equation (x2 + 16)2 + 80x2 = 18x(x2 + 16).

  1. 0

  2. 1

  3. 2

  4. 3

  5. 4
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let y = x^2 + 16. The equation is y^2 + 80x^2 = 18xy. Substitute y = x^2 + 16: (x^2 + 16)^2 + 80x^2 = 18x(x^2 + 16). x^4 + 32x^2 + 256 + 80x^2 = 18x^3 + 288x. x^4 - 18x^3 + 112x^2 - 288x + 256 = 0. Testing roots: x=4 gives 256 - 1152 + 1792 - 1152 + 256 = 0. Dividing by (x-4)^2 gives (x-4)^2(x^2 - 10x + 16) = 0, which is (x-4)^2(x-8)(x-2) = 0. The distinct roots are 2, 4, 8.

AI explanation

Let u = x^2 + 16 and rewrite the equation as u^2 + 80x^2 = 18xu, which rearranges to u^2 - 18xu + 80x^2 = 0. Factoring this quadratic in terms of u gives (u - 10x)(u - 8x) = 0, meaning u = 10x or u = 8x; substituting x^2 + 16 back for u yields x^2 + 16 = 10x and x^2 + 16 = 8x. The first equation becomes x^2 - 10x + 16 = 0 with roots 2 and 8, and the second becomes x^2 - 8x + 16 = 0 with a double root of 4. The distinct integral roots are therefore 2, 4, and 8, giving a total count of 3.

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