Question

Find the roots of the following equation, if they exist, by applying the quadratic formula:

`x^2 - (sqrt(3) + 1)x + sqrt(3) = 0`

Answer 1

Given: `x^2 - (sqrt(3) + 1)x + sqrt(3) = 0`

Step-wise calculation:

1. Compare with ax² + bx + c = 0:

a = 1, b = `-(sqrt(3) + 1)`, c = `sqrt(3)`

2. By the quadratic formula `x = (-b ± sqrt(b^2 - 4ac))/(2a)`.

3. Compute the discriminant:

`b^2 - 4ac = (sqrt(3) + 1)^2 - 4 xx 1 xx sqrt(3)` 

= `(3 + 2sqrt(3) + 1) - 4sqrt(3)`

= `4 - 2sqrt(3)`

4. Note `(sqrt(3) - 1)^2`

= `3 + 1 - 2sqrt(3)`

= `4 - 2sqrt(3)` 

So, `sqrt(b^2 - 4ac) = sqrt(3) - 1`.

5. Now `x = ((sqrt(3) + 1) ± (sqrt(3) - 1))/2`.

For +: `x = (2sqrt(3))/2 = sqrt(3)`. 

For −: `x = (2)/2 = 1`.

The roots are x = `sqrt(3)` and x = 1.