Algebra

sqrt(x) = x - 2

Step-by-step solution with explanation

Final Answer

(x = 1 is extraneous and rejected)

Step-by-step solution

1

Write down the equation clearly

We start with the given equation. Note that the square root requires x to be non-negative, so x ≥ 0. Also, since √x ≥ 0, we need x - 2 ≥ 0, meaning x ≥ 2.
2

Square both sides to remove radical

Squaring both sides eliminates the square root. We must check answers at the end because squaring can introduce false (extraneous) solutions.
3

Expand and simplify the equation

The left side becomes x. The right side expands using FOIL: .
4

Rearrange into standard quadratic form

Move x to the right side by subtracting x from both sides. Now we have a standard quadratic equation set equal to zero.
5

Factor the quadratic equation

We look for two numbers that multiply to 4 and add to -5. Those numbers are -1 and -4, so the factored form is (x-1)(x-4) = 0.
6

Solve for both possible values of x

Set each factor equal to zero: x - 1 = 0 gives x = 1, and x - 4 = 0 gives x = 4. These are our two candidate solutions.
7

Check both solutions in the original equation

Plug x = 1 back in: √1 = 1, but 1 - 2 = -1. Since 1 ≠ -1, x = 1 is extraneous and rejected. Plug x = 4 back in: √4 = 2, and 4 - 2 = 2. Since 2 = 2, x = 4 is valid.

Understanding this problem

Learning Insight

Squaring both sides is a powerful tool, but it can create extraneous solutions because squaring makes negative numbers positive. For example, √x = -1 has no real solution, but squaring gives x = 1, which looks like an answer. Always verify every solution in the original equation when squaring is involved.

Quick Tip

Before solving, notice that √x ≥ 0 always. So the right side (x - 2) must also be ≥ 0, meaning x ≥ 2. This alone tells you x = 1 cannot work, saving you time on the check.

Common Mistake

Students often forget to check solutions back in the original equation after squaring, and they incorrectly report x = 1 as a valid answer. Since √1 = 1 and 1 - 2 = -1 are not equal, x = 1 must be rejected as extraneous.