Final answer:
The equation √(x+3) - 4 = x - 1 has no non-extraneous solutions, as the potential solutions x = -2 and x = -3 do not satisfy the original equation when checked.
Step-by-step explanation:
The student is asking to find the non-extraneous solutions for the equation involving a square root: √(x+3) - 4 = x - 1. To solve for x, we first isolate the square root on one side by adding 4 to both sides, resulting in √(x+3) = x + 3. Squaring both sides gives us x + 3 = (x + 3)², which simplifies to x + 3 = x² + 6x + 9. Rearranging into standard quadratic form, we get x² + 5x + 6 = 0, which factors into (x + 2)(x + 3) = 0. This gives us two possible solutions for x: -2 and -3. However, we must check these solutions in the original equation to ensure they are not extraneous. Substituting x = -2 into the original equation, we find that √(-2+3) - 4 does not equal -2 - 1, thus x = -2 is an extraneous solution. Checking x = -3, we find √(-3+3) - 4 = -4, which also does not equal -3 - 1, meaning x = -3 is also an extraneous solution. Therefore, there are no non-extraneous solutions to the given equation.