Respuesta :
Answer:
Option 3 - The given equation is a prime.
Step-by-step explanation:
Given : Quadratic equation [tex]3x^2+x+7=0[/tex]
To find : The factors of given equation.
Solution : To factories the given equation [tex]3x^2+x+7=0[/tex]
We will apply discriminant method
General form - [tex]ax^2+bx+c=0[/tex]
[tex]D=b^2-4ac[/tex]
Solution is [tex]x=\frac{-b\pm\sqrt{D}}{2a}[/tex]
Equation is [tex]3x^2+x+7=0[/tex]
where a=3 , b=1, c=7
[tex]D=b^2-4ac[/tex]
[tex]D=(1)^2-4(3)(7)[/tex]
[tex]D=1-84[/tex]
[tex]D=-83[/tex]
Solution is [tex]x=\frac{-b\pm\sqrt{D}}{2a}[/tex]
[tex]x=\frac{-1\pm\sqrt{-83}}{2(3)}[/tex]
[tex]x=\frac{-1\pm\sqrt{83}i}{6}[/tex]
[tex]x=\frac{-1+\sqrt{83}i}{6},\frac{-1-\sqrt{83}i}{6}[/tex]
Therefore, The factors are
[tex][x+(\frac{-1+\sqrt{83}i}{6})][x-(\frac{-1-\sqrt{83}i}{6})][/tex]
So, 1,2,4 are not the options.
Now, check for prime
In a quadratic equation if [tex]D=\sqrt{b^2-4ac}[/tex] form a complete square then it is not prime and vice-versa.
In this question [tex]D=\sqrt{-83}[/tex] does not make a perfect square.
Therefore, It is a prime.
Option 3 is correct.
