What is the value of x?

Target question: What is the value of \(x\)?

Statement 1: \(\sqrt{2x + 2} −\sqrt{x − 3} = 2\)
Since we'll want to square both sides of the equation, let's make our calculations a little bit easier first by doing the following...

Add \(\sqrt{x − 3}\) to both sides of the equation to get: \(\sqrt{2x + 2} = 2 + \sqrt{x − 3}\)

Square both sides of the equation: \((\sqrt{2x + 2})^2 = (2 + \sqrt{x − 3})^2\)

Expand and simplify: \(2x + 2 = 4 + 4\sqrt{x − 3} + (x - 3)\)

Simplify the right side: : \(2x + 2 = 1 + 4\sqrt{x − 3} + x\)

Subtract \(1\) from both sides: : \(2x + 1 = 4\sqrt{x − 3} + x\)

Subtract \(x\) from both sides: : \(x + 1 = 4\sqrt{x − 3}\)

Square both sides: \((x + 1)^2 = (4\sqrt{x − 3})^2\)

Simplify: \(x^2 + 2x + 1 = 16(x - 3)\)

Expand the right side: \(x^2 + 2x + 1 = 16x - 48\)

Set the quadratic equation equal to zero: \(x^2 - 14x + 49 = 0\)

Factor: \((x - 7)(x - 7) = 0\)

So, it must be the case that \(x = 7\)

Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: \(x > 0\)
Clearly NOT SUFFICIENT

Answer: A

Cheers,
Brent