Solve (x-√7)(x+√7) = x²+7x+7: Square Root Equation Challenge

Solve the following equation:

_{$(x-\sqrt{7})(x+\sqrt{7})=x^2+7x+7$}

To solve this problem, we'll follow these steps:

• Step 1: Simplify the equation using the difference of squares.
• Step 2: Rearrange to form a suitable quadratic equation.
• Step 3: Solve the quadratic equation to find the value of $x$.

Now, let's work through each step:

Step 1: Simplify the left side using the difference of squares formula:

$(x - \sqrt{7})(x + \sqrt{7}) = x^2 - (\sqrt{7})^2 = x^2 - 7$

Step 2: Set the expressions equal and form a quadratic:

$x^2 - 7 = x^2 + 7x + 7$

Subtract $x^2$ from both sides:

$-7 = 7x + 7$

Rearrange the equation to isolate $x$:

$7x + 7 = -7$

Subtract 7 from both sides:

$7x = -14$

Divide both sides by 7:

$x = -2$

Therefore, the solution to the equation is $x = -2$.