Multiple choice

The equations x2 + rx + 64 = 0 and x2 - 8x + r = 0, where r > 0, have real roots. Then r satisfies the equation:

  1. r2 - 15r + 8 = 0

  2. r2 - 14r - 30 = 0

  3. r2 - 13r - 48 = 0

  4. r2 - 12r - 56 = 0
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For real roots, the discriminant must be >= 0. For x^2 + rx + 64 = 0, r^2 - 4(64) >= 0 => r^2 >= 256 => r >= 16. For x^2 - 8x + r = 0, 64 - 4r >= 0 => r <= 16. Thus, r must be 16. Substituting r=16 into the options, only C works: 16^2 - 13(16) - 48 = 256 - 208 - 48 = 0.

AI explanation

For the equations x2 + rx + 64 = 0 and x2 - 8x + r = 0 to have real roots, their discriminants must be non-negative, meaning r2 - 256 >= 0 and 64 - 4r >= 0. This gives r >= 16 or r <= -16, and r <= 16, so the only possible positive value is the boundary r = 16. Substituting r = 16 into the given options, option C yields (16)2 - 13(16) - 48 = 256 - 208 - 48 = 0. Thus, r satisfies the equation r2 - 13r - 48 = 0.

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