Multiple choice

The equation $x + 5-\sqrt {x + 5} = 6$ has

  1. only one root

  2. exactly two roots

  3. a infinitely many roots

  4. no root
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let u = sqrt(x + 5). Then u^2 - u - 6 = 0. (u - 3)(u + 2) = 0. Since u must be >= 0, u = 3. Thus sqrt(x + 5) = 3, x + 5 = 9, x = 4. Only one root exists.

AI explanation

Substitute y equal to the square root of (x + 5) into the equation, which transforms it into y squared minus y minus 1 equals 0. Using the quadratic formula, the roots for y are (1 plus or minus the square root of 5) divided by 2, which are approximately 1.618 and negative 0.618. Because the substitution variable y represents a principal square root, it cannot be negative, leaving only y equal to 1.618 as a valid solution, which gives exactly one corresponding real root for x. Thus, the equation has only one root.

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