This example approximates the square root of using algebra.

Start by representing with the variable .

Multiply both sides by

Substitute from step 1 in for on the right side.

Add to both sides.

Factor from the left side.

Divide both sides by .

Simplify the left side and split the right side fraction into two parts.

Simplify again.

Substitute from step 1 on the left side.

Recursively substitute the expression from step 8 in for .

Combine like terms.

Recursively substitute the expression from step 8 in for .

Combine like terms.

Repeat the recursive substition for more accuracy. To calculate an approximate value substitute into the expression. This is shown below.

Recursively substituting six times gives us an approximation of that is accurate for 5-digits. This can be repeated for more and more accuracy.