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:
- \(a = 4\),
- \(b = 1\),
- \(c = 1\).
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