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.
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Start with the equation: \[ \sqrt{a} - \sqrt{b} = \sqrt{c}. \]
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Square both sides: \[ (\sqrt{a} - \sqrt{b})^2 = \sqrt{c}^2, \] which simplifies to: \[ a - 2\sqrt{a}\sqrt{b} + b = c. \]
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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:
- \(9, 4, 1\).
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