Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
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]
[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]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.