Question

Discuss the nature of the roots of the following equation without actually solving it:

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

Answer 1

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

Step-wise calculation:

1. Compare with ax² + bx + c = 0: 

a = 1, b = `2sqrt(3)`, c = –1

2. Discriminant Δ = b² – 4ac 

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

= 12 + 4

= 16

Since Δ > 0, the roots are real and unequal.

3. Use the quadratic formula:

`x = (-b ± sqrt(Δ))/(2a)` 

= `(-2sqrt(3) ± 4)/2` 

= `-sqrt(3) ± 2` 

So the two roots are `2 - sqrt(3)` and `-2 - sqrt(3)`.

4. `sqrt(3)` is irrational, and adding or subtracting a rational number (±2) to an irrational number yields an irrational number; therefore each root is irrational. The two signs give different values their difference is 4, so they are unequal.

The equation has two distinct (unequal) real roots and each root is irrational specifically `2 - sqrt(3)` and `-2 - sqrt(3)`.