Question

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

x² – 4x – 1 = 0

Answer 1

Given: x² – 4x – 1 = 0

Step-wise calculation:

1. Identify coefficients:

a = 1, b = –4, c = –1

2. Quadratic formula:

`x = (-b ± sqrt(b^2 - 4ac))/(2a)`

3. Compute discriminant:

D = b² – 4ac 

= (–4)² – 4(1)(–1) 

= 16 + 4

= 20

4. `sqrt(D) = sqrt(20)` 

= `2sqrt(5)`

5. Substitute:

`x = (-(-4) ± 2sqrt(5))/(2 xx 1)` 

= `(4 ± 2sqrt(5))/2` 

= `2 ± sqrt(5)`

The roots are `x = 2 ± sqrt(5)` and `x = 2 - sqrt(5)` both real and irrational.