Factor completely 3x2 + x + 7.
(3x + 1)(x + 7)
(3x + 7)(x + 1)
Prime
(3x + 4)(x + 3)
Answer:
Option 3 - The given equation is a prime.
Step-by-step explanation:
Given : Quadratic equation \(3x^2+x+7=0\)
To find : The factors of given equation.
Solution : To factories the given equation \(3x^2+x+7=0\)
We will apply discriminant method
General form - \(ax^2+bx+c=0\)
\(D=b^2-4ac\)
Solution is \(x=\frac{-b\pm\sqrt{D}}{2a}\)
Equation is \(3x^2+x+7=0\)
where a=3 , b=1, c=7
\(D=b^2-4ac\)
\(D=(1)^2-4(3)(7)\)
\(D=1-84\)
\(D=-83\)
Solution is \(x=\frac{-b\pm\sqrt{D}}{2a}\)
\(x=\frac{-1\pm\sqrt{-83}}{2(3)}\)
\(x=\frac{-1\pm\sqrt{83}i}{6}\)
\(x=\frac{-1+\sqrt{83}i}{6},\frac{-1-\sqrt{83}i}{6}\)
Therefore, The factors are
\([x+(\frac{-1+\sqrt{83}i}{6})][x-(\frac{-1-\sqrt{83}i}{6})]\)
So, 1,2,4 are not the options.
Now, check for prime
In a quadratic equation if \(D=\sqrt{b^2-4ac}\) form a complete square then it is not prime and vice-versa.
In this question \(D=\sqrt{-83}\) does not make a perfect square.
Therefore, It is a prime.
Option 3 is correct.