Identify the extraneous solution when solving √x+2= x
To solve the equation √x + 2 = x, you can follow these steps:
1. Subtract 2 from both sides of the equation: √x = x - 2.
2. Square both sides of the equation to eliminate the square root: (√x)² = (x - 2)².
This gives us x = (x - 2)².
Now, let's solve for x:
Expanding the right side of the equation, we have x = x² - 4x + 4.
Rearranging the equation, we get:
x² - 5x + 4 = 0.
Factoring the quadratic equation, we have (x - 4)(x - 1) = 0.
Setting each factor equal to zero, we get two possible solutions:
x - 4 = 0 → x = 4.
x - 1 = 0 → x = 1.
However, when we substitute x = 4 and x = 1 back into the original equation √x + 2 = x, only x = 4 satisfies the equation:
When x = 4, √4 + 2 = 4 → 2 + 2 = 4, which is true.
When x = 1, √1 + 2 = 1 → 1 + 2 = 1, which is not true.
Thus, the solution x = 1 is the extraneous solution since it does not satisfy the original equation.