Solve the following equations :
`(x-5)^(2)-(x+2)^(2)=-2`

Answer

To solve the equation \((x-5)^{2} - (x+2)^{2} = -2\), we will follow these steps:

### Step 1: Expand both squares
Using the formula for the square of a binomial, we can expand \((x-5)^{2}\) and \((x+2)^{2}\).

\[ (x-5)^{2} = x^{2} - 10x + 25 \]
\[ (x+2)^{2} = x^{2} + 4x + 4 \]

### Step 2: Substitute the expansions into the equation
Now, substitute these expansions back into the original equation:

\[ (x^{2} - 10x + 25) - (x^{2} + 4x + 4) = -2 \]

### Step 3: Simplify the equation
Distributing the negative sign and combining like terms:

\[ x^{2} - 10x + 25 - x^{2} - 4x - 4 = -2 \]

The \(x^{2}\) terms cancel out:

\[ -10x - 4x + 25 - 4 = -2 \]

Combine the \(x\) terms and the constants:

\[ -14x + 21 = -2 \]

### Step 4: Isolate the variable \(x\)
Now, we will isolate \(x\) by moving the constant to the other side:

\[ -14x = -2 - 21 \]
\[ -14x = -23 \]

Now, divide both sides by -14:

\[ x = \frac{-23}{-14} = \frac{23}{14} \]

### Final Solution
Thus, the solution to the equation is:

\[ x = \frac{23}{14} \]

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