Question

Factor completely 3x2 + x + 7.

(3x + 1)(x + 7)
(3x + 7)(x + 1)
Prime
(3x + 4)(x + 3)

Answer 1

Option 3 - The given equation is a prime.

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=(-b\pm√(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=(-b\pm√(D))/(2a)`

`x=(-1\pm√(-83))/(2(3))`

`x=(-1\pm√(83)i)/(6)`

`x=(-1+√(83)i)/(6),(-1-√(83)i)/(6)`

Therefore, The factors are

`[x+((-1+√(83)i)/(6))][x-((-1-√(83)i)/(6))]`

So, 1,2,4 are not the options.

Now, check for prime

In a quadratic equation if
`D=√(b^2-4ac)` form a complete square then it is not prime and vice-versa.

In this question
`D=√(-83)` does not make a perfect square.

Therefore, It is a prime.

Option 3 is correct.