If `2sqrtx=x-3`, which of the following is the solution set for x?

Options

A. {-1,9}
B. {1,-9}
C. {9}
D. {1,9}

Answer

To solve the equation \( 2\sqrt{x} = x - 3 \), we will follow these steps:

### Step 1: Square both sides
We start with the equation:
\[ 2\sqrt{x} = x - 3 \]
Squaring both sides, we get:
\[ (2\sqrt{x})^2 = (x - 3)^2 \]
This simplifies to:
\[ 4x = x^2 - 6x + 9 \]

### Step 2: Rearrange the equation
Next, we rearrange the equation to bring all terms to one side:
\[ 4x - x^2 + 6x - 9 = 0 \]
This simplifies to:
\[ -x^2 + 10x - 9 = 0 \]
Multiplying through by -1 gives us:
\[ x^2 - 10x + 9 = 0 \]

### Step 3: Factor the quadratic equation
Now we will factor the quadratic equation:
\[ x^2 - 10x + 9 = 0 \]
We look for two numbers that multiply to \(9\) and add up to \(-10\). The numbers are \(-1\) and \(-9\):
\[ (x - 1)(x - 9) = 0 \]

### Step 4: Solve for x
Setting each factor to zero gives us the possible solutions:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 9 = 0 \quad \Rightarrow \quad x = 9 \]

### Step 5: Check for extraneous solutions
We need to check both solutions in the original equation to see if they are valid.

**For \(x = 1\)**:
\[ 2\sqrt{1} = 1 - 3 \quad \Rightarrow \quad 2 \neq -2 \quad \text{(not valid)} \]

**For \(x = 9\)**:
\[ 2\sqrt{9} = 9 - 3 \quad \Rightarrow \quad 6 = 6 \quad \text{(valid)} \]

### Conclusion
The only solution that satisfies the original equation is:
\[ \{9\} \]