The roots of quadratic equation x² - 4x - 1 = 0 are,

Correct option is A

Concept/Formula Used:

For a quadratic equation of the form $$ax^2 + bx + c = 0$$​ , the roots are given by the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$​

where:
- a is the coefficient of $$x^2$$​ ,
- b is the coefficient of x ,
- c is the constant term.

Given:

The equation is:
$$x^2 - 4x - 1 = 0$$​
Here:
$$a = 1, \ b = -4, \ c = -1$$​

Formula Used:

Using the quadratic formula:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$​

Solution:

Substitute a = 1 , b = -4 , and c = -1 into the formula:

1. Compute $$b^2 - 4ac$$​ :

b^2 - 4ac = (-4)^2 - 4(1)(-1) = 16 + 4 = 20


2. Substitute into the quadratic formula:

$$x = \frac{-(-4) \pm \sqrt{20}}{2(1)}$$​


$$x = \frac{4 \pm \sqrt{20}}{2}$$​


3. Simplify $$\sqrt{20}$$​ :

$$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$$​


4. Simplify the expression:

$$x = \frac{4 \pm 2\sqrt{5}}{2}$$​


$$x = 2 \pm \sqrt{5}$$​

Correct Answer:

$$(A) 2 \pm \sqrt{5}$$​