First, let’s address the problem systematically to solve the quadratic equation:
[tex]\[ 11x^2 - 4x - 1 = 0 \][/tex]
We need to find the values of [tex]\( x \)[/tex] using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 11 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = -1 \)[/tex].
1. Calculate the Discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]
Substituting the values:
[tex]\[ \text{Discriminant} = (-4)^2 - 4 \times 11 \times (-1) \][/tex]
[tex]\[ \text{Discriminant} = 16 + 44 \][/tex]
[tex]\[ \text{Discriminant} = 60 \][/tex]
2. Apply the Quadratic Formula:
Since the discriminant is positive, there will be two real solutions.
[tex]\[ x = \frac{-(-4) \pm \sqrt{60}}{2 \times 11} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{60}}{22} \][/tex]
3. Simplify:
[tex]\[ \sqrt{60} = \sqrt{4 \times 15} = 2\sqrt{15} \][/tex]
So:
[tex]\[ x = \frac{4 \pm 2\sqrt{15}}{22} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{15}}{11} \][/tex]
Therefore, the solutions to the quadratic equation [tex]\( 11x^2 - 4x - 1 = 0 \)[/tex] are:
[tex]\[ x = \frac{2}{11} \pm \frac{\sqrt{15}}{11} \][/tex]
Hence, the correct choice is:
[tex]\[ \boxed{\frac{2}{11} \pm \frac{\sqrt{15}}{11}} \][/tex]
