Question

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

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. Let's move \(-\sqrt{b}\) to the right side: \(\sqrt{b + 20}=\sqrt{b}+5\). Here, we added the radical term \(\sqrt{b}\) to both sides (so we add a radical term to both sides).

Step2: Square both sides

Now, square both sides to eliminate the square root on the left: \((\sqrt{b + 20})^2 = (\sqrt{b}+5)^2\), which simplifies to \(b + 20=b + 10\sqrt{b}+25\). Then, we can simplify this equation further, but notice that we squared both sides only once so far? Wait, no, actually, after the first squaring, we still have a radical (\(\sqrt{b}\)) left. Wait, no, let's re - examine. Wait, the original equation has two radical terms. Wait, no, the original equation is \(\sqrt{b + 20}-\sqrt{b}=5\). When we move \(\sqrt{b}\) to the right, we get \(\sqrt{b + 20}=\sqrt{b}+5\), then square both sides. After that, we have a linear term and a radical? Wait, no, \( (\sqrt{b}+5)^2=b + 10\sqrt{b}+25\), so we still have a radical. So we need to isolate the remaining radical and square again? Wait, no, in this case, let's check the options. The options are about adding a radical term and squaring. Let's see the process:

First, we add the radical term \(\sqrt{b}\) to both sides (so we add a radical term to both sides) to get \(\sqrt{b + 20}=\sqrt{b}+5\), then we square both sides. But wait, after squaring, we still have a radical? Wait, no, \( (\sqrt{b}+5)^2=b + 10\sqrt{b}+25\), so \(\sqrt{b + 20}=\sqrt{b}+5\) squared gives \(b + 20=b + 10\sqrt{b}+25\), then we can simplify to \(20 = 10\sqrt{b}+25\), then \(- 5=10\sqrt{b}\), which is not possible. Wait, maybe I made a mistake. Wait, the correct process: Let's take the original equation \(\sqrt{b + 20}-\sqrt{b}=5\). Let's add \(\sqrt{b}\) to both sides (add a radical term to both sides) to get \(\sqrt{b + 20}=\sqrt{b}+5\). Then square both sides: \(b + 20=(\sqrt{b}+5)^2=b + 10\sqrt{b}+25\). Then subtract \(b\) and 25 from both sides: \(- 5 = 10\sqrt{b}\), which is wrong. Wait, maybe the equation is \(\sqrt{b + 20}-\sqrt{b}=5\), let's check with \(b = 4\): \(\sqrt{24}-\sqrt{4}=\sqrt{24}-2\approx4.9 - 2 = 2.9 eq5\). \(b = 9\): \(\sqrt{29}-\sqrt{9}\approx5.385 - 3 = 2.385 eq5\). \(b = 16\): \(\sqrt{36}-\sqrt{16}=6 - 4 = 2 eq5\). \(b = 25\): \(\sqrt{45}-\sqrt{25}\approx6.708 - 5 = 1.708 eq5\). \(b = 0\): \(\sqrt{20}-0\approx4.47 eq5\). Wait, maybe the equation is \(\sqrt{b + 20}+\sqrt{b}=5\)? No, the user wrote \(\sqrt{b + 20}-\sqrt{b}=5\). Anyway, back to the options. The options are:

1. Add a constant term to both sides and square both sides only once.

2. Add a constant term to both sides and square both sides twice.

3. Add a radical term to both sides and square both sides twice.

4. Add a radical term to both sides and square both sides only once.

Wait, in our process, we added a radical term (\(\sqrt{b}\)) to both sides, and then we squared once, but we still had a radical? No, wait, no, in the equation \(\sqrt{b + 20}=\sqrt{b}+5\), when we square, we get \(b + 20=b + 10\sqrt{b}+25\), so we still have a radical. So we need to isolate the radical and square again? But the options don't have that. Wait, maybe the equation is different. Wait, maybe the equation is \(\sqrt{b + 20}-\sqrt{b}=5\), let's do it correctly. Let's let \(x=\sqrt{b}\), then the equation becomes \(\sqrt{x^{2}+20}-x = 5\), then \(\sqrt{x^{2}+20}=x + 5\), square both sides: \(x^{2}+20=x^{2}+10x + 25\), then \(20 = 10x+25\), \(10x=-5\), \(x =-\frac{1}{2}\), but \(x=\sqrt{b}\geq0\), so no solution. But the question is about the process. The process is: we add a radical term (\(\sqrt{b}\)) to both sides (so add a radical term to both sides) and then square both sides. But in the options, the fourth option is "Add a radical term to both sides and square both sides only once". But in reality, we might need to square twice, but maybe in this case, the first squaring is the main step. Wait, the options:

Option 4: Add a radical term to both sides and square both sides only once.

Let's check the options again. The options are:

1. Add a constant term to both sides and square both sides only once.

2. Add a constant term to both sides and square both sides twice.

3. Add a radical term to both sides and square both sides twice.

4. Add a radical term to both sides and square both sides only once.

In the process of solving \(\sqrt{b + 20}-\sqrt{b}=5\), we first add the radical term \(\sqrt{b}\) to both sides (add a radical term to both sides) and then square both sides (only once, even though we might have made a mistake in the equation's solvability, the process is about the number of times we square and what we add). So the correct option is the fourth one: "Add a radical term to both sides and square both sides only once."

Answer:

Add a radical term to both sides and square both sides only once. (The option corresponding to this description, assuming it's the fourth option in the list)