Question

which statement describes a process to solve \\(\sqrt{b + 20} - \sqrt{b} = 5\\)?
\\(\bigcirc\\) add a radical term to both sides and square both sides only once.
\\(\bigcirc\\) add a constant term to both sides and square both sides only once.
\\(\bigcirc\\) add a radical term to both sides and square both sides twice.
\\(\bigcirc\\) add a constant term to both sides and square both sides twice.

Explanation:

Step1: Analyze the equation

The equation is \(\sqrt{b + 20}-\sqrt{b}=5\). To solve it, we first need to isolate one of the radical terms. So we add \(\sqrt{b}\) (a radical term) to both sides to get \(\sqrt{b + 20}=5+\sqrt{b}\).

Step2: Square both sides for the first time

Squaring both sides of \(\sqrt{b + 20}=5+\sqrt{b}\), we have \((\sqrt{b + 20})^2=(5+\sqrt{b})^2\), which simplifies to \(b + 20 = 25+10\sqrt{b}+b\).

Step3: Simplify and isolate the remaining radical

Subtract \(b\) and 25 from both sides: \(b + 20 - b-25=10\sqrt{b}\), so \(- 5 = 10\sqrt{b}\), then divide by 10: \(-\frac{1}{2}=\sqrt{b}\). Wait, but we still have a radical here, so we need to square both sides again to eliminate the square root. Squaring both sides of \(-\frac{1}{2}=\sqrt{b}\) (even though this will lead to a contradiction as square root can't be negative, but the process requires squaring twice). So we added a radical term first and then squared twice.

Answer:

Add a radical term to both sides and square both sides twice. (The option: Add a radical term to both sides and square both sides twice.)