Complete the expression subtracting the radicals.

?sqrt?-?sqrt?=?sqrt?
find the value of all ?s

1 answer

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:

  • \(9, 4, 1\).