Bunuel
What is the value of \(x\)?
(1) \(\sqrt{2x + 2} −\sqrt{x − 3} = 2\)
(2) \(x > 0\)
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