Question

Complete the expression subtracting the radicals.
?sqrt?-?sqrt?=?sqrt?
find the value of all ?s

Answer 1

To complete the expression of the form

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

we'll solve for \(a\), \(b\), and \(c\). By squaring both sides of the equation, we can manipulate the equation to find the desired values.

1. Start with the equation:
\[
\sqrt{a} - \sqrt{b} = \sqrt{c}.
\]

2. Square both sides:
\[
(\sqrt{a} - \sqrt{b})^2 = \sqrt{c}^2,
\]
which simplifies to:
\[
a - 2\sqrt{a}\sqrt{b} + b = c.
\]

3. Rearranging gives us:
\[
c = a + b - 2\sqrt{a}\sqrt{b}.
\]

To find values of \(a\), \(b\), and \(c\) that satisfy this relationship, let's choose specific values for \(a\) and \(b\).

For instance, let:
- \(a = 9\),
- \(b = 4\).

So now calculate \(c\):

\[
c = 9 + 4 - 2\sqrt{9}\sqrt{4} = 9 + 4 - 2 \cdot 3 \cdot 2 = 9 + 4 - 12 = 1.
\]

Thus:
\[
\sqrt{9} - \sqrt{4} = \sqrt{1}.
\]
This is valid because:

\[
3 - 2 = 1.
\]

The final values are:
- \(a = 9\),
- \(b = 4\),
- \(c = 1\).

Thus, the complete expression is:

\[
\sqrt{9} - \sqrt{4} = \sqrt{1}.
\]

So the values of all \( ? \)s can be filled as follows:
- \(9, 4, 1\).