For what value of p is `(x-2)(x+2)=x(x-p)`?

Options

A. `-4`
B. `0`
C. `(2)/(x)`
D. `(4)/(x)`

Answer

To solve the equation \((x-2)(x+2) = x(x-p)\), we will follow these steps:

### Step 1: Expand both sides of the equation.
First, we will expand the left-hand side:
\[ (x-2)(x+2) = x^2 - 4 \]

Now, expand the right-hand side:
\[ x(x-p) = x^2 - px \]

### Step 2: Set the expanded forms equal to each other.
Now we can set the two expanded forms equal to each other:
\[ x^2 - 4 = x^2 - px \]

### Step 3: Simplify the equation.
We can subtract \(x^2\) from both sides:
\[ -4 = -px \]

### Step 4: Solve for \(p\).
To isolate \(p\), we can rearrange the equation:
\[ px = 4 \]
Now, divide both sides by \(x\) (assuming \(x \neq 0\)):
\[ p = \frac{4}{x} \]

### Conclusion:
The value of \(p\) is \(\frac{4}{x}\).

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