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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).
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