Find the value (s) of x , if the distance between the points A(0,0) and B(x,-4) is 5 units.

Answer

To find the value(s) of \( x \) such that the distance between the points \( A(0, 0) \) and \( B(x, -4) \) is 5 units, we can follow these steps:

### Step 1: Use the Distance Formula
The distance \( d \) between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In our case, \( A(0, 0) \) and \( B(x, -4) \). Thus, we can substitute \( x_1 = 0 \), \( y_1 = 0 \), \( x_2 = x \), and \( y_2 = -4 \).

### Step 2: Set Up the Equation
According to the distance formula:
\[ d = \sqrt{(x - 0)^2 + (-4 - 0)^2} \]
This simplifies to:
\[ d = \sqrt{x^2 + 16} \]
Since the distance is given as 5 units, we set up the equation:
\[ \sqrt{x^2 + 16} = 5 \]

### Step 3: Eliminate the Square Root
To eliminate the square root, we square both sides of the equation:
\[ (\sqrt{x^2 + 16})^2 = 5^2 \]
This simplifies to:
\[ x^2 + 16 = 25 \]

### Step 4: Solve for \( x^2 \)
Now, we can isolate \( x^2 \):
\[ x^2 = 25 - 16 \]
\[ x^2 = 9 \]

### Step 5: Find the Values of \( x \)
To find \( x \), we take the square root of both sides:
\[ x = \pm 3 \]
Thus, the possible values of \( x \) are:
\[ x = 3 \quad \text{and} \quad x = -3 \]

### Final Answer
The values of \( x \) are \( 3 \) and \( -3 \).

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