Question

What is the solution for the equation x-√x-1=3?

A. x = 2 only, because x = 5 is extraneous.
OB. No solution, because x = -5 and x = -2 are both extraneous.
OC. x = 5 only, because x = 2 is extraneous.
D. x = 2 or x = 5

Answer 1

Hi,

Answer : C. x = 5 only, because x = 2 is extraneous.

Explanation:

Let's solve the equation step by step:

`x-√(x-1) =3\\\Longrightarrow\ √(x-1) =3-x`

Now, square both sides of the equation to eliminate the square root:

`x-1=(3-x)^2\\x-1=9-6x+x^2\\`

Now, move all terms to one side to set the equation to zero:

`x^2 - 6x + 9 - x + 1 = 0\\`

Combine like terms:

`x^2 - 7x + 10 = 0\\`

Now, let's factor the quadratic equation:

`x^2-7x+10=0\\x^2-2x-5x+10=0\\x(x-2)-5(x-2)=0\\(x-2)(x-5)=0\\`

Now, set each factor equal to zero:

x - 2 = 0

x = 2

x - 5 = 0

x = 5

So, the solutions for the equation are x = 2 and x = 5.

Now, let's check whether any of these solutions are extraneous. We started with a square root, so we need to make sure the values we found make sense within the context of the original equation:

For x = 2:

√(2 - 1) = √1 = 1

2 - 3 = -1

The left side is 1, and the right side is -1, so x = 2 is extraneous.

For x = 5:

√(5 - 1) = √4 = 2

5 - 3 = 2

Both sides are equal when x = 5, so x = 5 is a valid solution.

So, the correct answer is:

C. x = 5 only, because x = 2 is extraneous.