Question

If `x = 7 + 4sqrt(3)`, find the value of `sqrt(x) - 1/sqrt(x)`.

Answer 1

Given: `x = 7 + 4sqrt(3)`

Stepwise calculation:

1. First, find the value of `sqrt(x)`. 

Observe that `7 + 4sqrt(3)` can be expressed in the form `(a + bsqrt(3))^2 = a^2 + 2ab sqrt(3) + 3b^2`. 

We want a² + 3b² = 7, 2ab = 4.

2. From 2ab = 4, we get:

ab = 2

⇒ `b = 2/a`

3. Substitute into the first equation:

`a^2 + 3(2/a)^2 = 7`

`a^2 + 3 xx 4/a^2 = 7`

Multiply both sides by a² : a⁴ + 12 = 7a².

Rearranging: a⁴ – 7a² + 12 = 0.

Let y = a², the equation becomes y² – 7y + 12 = 0.

4. Solve quadratic:

`y = (7 +- sqrt(49 - 48))/2`

`y = (7 +- 1)/2`

Thus, y = 4 or y = 3.

5. Since a² = 4 or 3,

If a² = 4, then a = 2.

Using ab = 2, `b = 2/2 = 1`. 

Check if a² + 3b² = 4 + 3(1) = 7, correct.

If a² = 3, `a = sqrt(3)`. 

Then, `b = 2/sqrt(3) = (2sqrt(3))/3`, leading to a rational b. 

But for simplicity, choose a = 2, b = 1.

6. So, `sqrt(x) = 2 + sqrt(3)`.

7. Calculate `\(1/sqrt(x))`:

`1/(2 + sqrt(3))`

= `(2 - sqrt(3))/((2 + sqrt(3))(2 - sqrt(3))`

= `(2 - sqrt(3))/(4 - 3)`

= `2 - sqrt(3)`

8. Finally, compute:

`sqrt(x) - 1/sqrt(x)`

= `(2 + sqrt(3)) - (2 - sqrt(3))`

= `2 + sqrt(3) - 2 + sqrt(3)`

= `2sqrt(3)`