Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Join our platform to connect with experts ready to provide accurate answers to your questions in various fields. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
Para resolver la ecuación cuadrática [tex]\(7x^2 - 3x - 2 = 0\)[/tex], comenzamos identificando los coeficientes de la ecuación cuadrática de la forma [tex]\(ax^2 + bx + c = 0\)[/tex]:
- [tex]\(a = 7\)[/tex]
- [tex]\(b = -3\)[/tex]
- [tex]\(c = -2\)[/tex]
Para encontrar las raíces de la ecuación, utilizamos la fórmula cuadrática:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Primero, calculemos el discriminante ([tex]\(\Delta\)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Sustituimos los valores de [tex]\(a\)[/tex], [tex]\(b\)[/tex], y [tex]\(c\)[/tex]:
[tex]\[ \Delta = (-3)^2 - 4 \times 7 \times (-2) \][/tex]
[tex]\[ \Delta = 9 + 56 \][/tex]
[tex]\[ \Delta = 65 \][/tex]
Ahora utilizamos el valor del discriminante para encontrar las dos raíces:
[tex]\[ x_{1} = \frac{-b + \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x_{1} = \frac{-(-3) + \sqrt{65}}{2 \times 7} \][/tex]
[tex]\[ x_{1} = \frac{3 + \sqrt{65}}{14} \][/tex]
[tex]\[ x_{2} = \frac{-b - \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x_{2} = \frac{-(-3) - \sqrt{65}}{2 \times 7} \][/tex]
[tex]\[ x_{2} = \frac{3 - \sqrt{65}}{14} \][/tex]
Las raíces son:
[tex]\[ x_{1} = \frac{3 + \sqrt{65}}{14} \quad \text{y} \quad x_{2} = \frac{3 - \sqrt{65}}{14} \][/tex]
Entre estas dos raíces, la menor es [tex]\( \frac{3 - \sqrt{65}}{14} \)[/tex].
Finalmente, la respuesta correcta es:
[tex]\[ \boxed{\frac{3 - \sqrt{65}}{14}} \][/tex]
Por lo tanto, la opción correcta es la (e).
- [tex]\(a = 7\)[/tex]
- [tex]\(b = -3\)[/tex]
- [tex]\(c = -2\)[/tex]
Para encontrar las raíces de la ecuación, utilizamos la fórmula cuadrática:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Primero, calculemos el discriminante ([tex]\(\Delta\)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Sustituimos los valores de [tex]\(a\)[/tex], [tex]\(b\)[/tex], y [tex]\(c\)[/tex]:
[tex]\[ \Delta = (-3)^2 - 4 \times 7 \times (-2) \][/tex]
[tex]\[ \Delta = 9 + 56 \][/tex]
[tex]\[ \Delta = 65 \][/tex]
Ahora utilizamos el valor del discriminante para encontrar las dos raíces:
[tex]\[ x_{1} = \frac{-b + \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x_{1} = \frac{-(-3) + \sqrt{65}}{2 \times 7} \][/tex]
[tex]\[ x_{1} = \frac{3 + \sqrt{65}}{14} \][/tex]
[tex]\[ x_{2} = \frac{-b - \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x_{2} = \frac{-(-3) - \sqrt{65}}{2 \times 7} \][/tex]
[tex]\[ x_{2} = \frac{3 - \sqrt{65}}{14} \][/tex]
Las raíces son:
[tex]\[ x_{1} = \frac{3 + \sqrt{65}}{14} \quad \text{y} \quad x_{2} = \frac{3 - \sqrt{65}}{14} \][/tex]
Entre estas dos raíces, la menor es [tex]\( \frac{3 - \sqrt{65}}{14} \)[/tex].
Finalmente, la respuesta correcta es:
[tex]\[ \boxed{\frac{3 - \sqrt{65}}{14}} \][/tex]
Por lo tanto, la opción correcta es la (e).
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.