If the distance between the point `(x,-1)` and `(-2,2)` is 5 , then find the possible values of x .

Answer

To solve the problem, we need to find the possible values of \( x \) such that the distance between the points \( (x, -1) \) and \( (-2, 2) \) is equal to 5.

### Step-by-Step Solution:

1. **Use the Distance Formula**:
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \]
Here, \( (x_1, y_1) = (x, -1) \) and \( (x_2, y_2) = (-2, 2) \). The distance is given as 5.

2. **Set Up the Equation**:
Substitute the points into the distance formula:
\[ 5 = \sqrt{(x - (-2))^2 + (-1 - 2)^2} \]
This simplifies to:
\[ 5 = \sqrt{(x + 2)^2 + (-3)^2} \]

3. **Square Both Sides**:
To eliminate the square root, square both sides of the equation:
\[ 25 = (x + 2)^2 + 9 \]

4. **Rearrange the Equation**:
Subtract 9 from both sides:
\[ 25 - 9 = (x + 2)^2 \]
\[ 16 = (x + 2)^2 \]

5. **Take the Square Root**:
Take the square root of both sides:
\[ \sqrt{16} = x + 2 \quad \text{or} \quad \sqrt{16} = -(x + 2) \]
This gives us two equations:
\[ 4 = x + 2 \quad \text{and} \quad -4 = x + 2 \]

6. **Solve for \( x \)**:
For the first equation:
\[ x + 2 = 4 \implies x = 4 - 2 = 2 \]
For the second equation:
\[ x + 2 = -4 \implies x = -4 - 2 = -6 \]

7. **Final Values**:
Therefore, the possible values of \( x \) are:
\[ x = 2 \quad \text{and} \quad x = -6 \]

### Conclusion:
The possible values of \( x \) are \( 2 \) and \( -6 \).

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