Multiple choice

Roots of the equation $x^{2}-6x-\sqrt{x^{2}-6x-3}=5$ are

  1. $1, 7$
  2. $-1, -7$
  3. $1$, $-1$
  4. $-1$, $7$
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D Correct answer
Explanation

Let y = sqrt(x^2 - 6x - 3). Then y^2 = x^2 - 6x - 3, so x^2 - 6x = y^2 + 3. The equation becomes y^2 + 3 - y = 5, or y^2 - y - 2 = 0. Solving gives y = 2 or y = -1 (discarded). x^2 - 6x - 3 = 4 leads to x^2 - 6x - 7 = 0, so (x-7)(x+1) = 0.

AI explanation

Substituting y = sqrt(x^2 - 6x - 3), the equation becomes y^2 + 3 - y = 5, which simplifies to y^2 - y - 2 = 0. Factoring this quadratic gives (y - 2)(y + 1) = 0, so y = 2 or y = -1; since y is a principal square root, y cannot be negative and must be 2. Setting sqrt(x^2 - 6x - 3) = 2 and squaring both sides gives x^2 - 6x - 3 = 4, which rearranges to x^2 - 6x - 7 = 0. Solving this yields the roots -1 and 7.

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