Question

Complete the expression subtracting the radicals.
?sqrt?-?sqrt?=?sqrt?
not all ?s are the same. find the value of all ?s

Answer 1

To complete the expression \(\sqrt{?} - \sqrt{?} = \sqrt{?}\), let's denote the values under the radicals as follows:

Let:
- \(\sqrt{a}\) be the first radical,
- \(\sqrt{b}\) be the second radical,
- \(\sqrt{c}\) be the result.

We want to form an equation based on the expression:

\[
\sqrt{a} - \sqrt{b} = \sqrt{c}
\]

To solve for \(a\), \(b\), and \(c\), we can first rearrange the equation to isolate the radicals on one side:

\[
\sqrt{a} = \sqrt{b} + \sqrt{c}
\]

Next, we square both sides to eliminate the radicals:

\[
a = (\sqrt{b} + \sqrt{c})^2
\]

Expanding the right side gives:

\[
a = b + c + 2\sqrt{bc}
\]

This means we will need to assign values to \(b\) and \(c\) such that \(a\) can be expressed in its square root form.

Let's assign values to \(b\) and \(c\). A simple choice is to let \(b = 1\) and \(c = 1\):

- Assign \(b = 1\) implies \(\sqrt{1} = 1\),
- Assign \(c = 1\) implies \(\sqrt{1} = 1\).

Now, we calculate \(a\):

\[
a = 1 + 1 + 2\sqrt{1 \cdot 1} = 1 + 1 + 2 = 4
\]

Thus, we have the values:
- \(a = 4\),
- \(b = 1\),
- \(c = 1\).

Now, substituting back:

\[
\sqrt{4} - \sqrt{1} = \sqrt{1}
\]

This becomes:

\[
2 - 1 = 1,
\]

which is correct. Therefore, the completed expression with all ?s filled is:

\[
\sqrt{4} - \sqrt{1} = \sqrt{1}.
\]

So, the final values are:
- \(a = 4\),
- \(b = 1\),
- \(c = 1\).