To determine the number of roots, including complex roots, of the polynomial [tex]\( 3x^2 + 4 \)[/tex], we'll employ the concept of the discriminant from the quadratic formula.
The general form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex].
In this particular case:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 0 \)[/tex]
- [tex]\( c = 4 \)[/tex]
For any quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], the discriminant ([tex]\( D \)[/tex]) is given by the formula:
[tex]\[ D = b^2 - 4ac \][/tex]
Substituting the given values:
[tex]\[ D = 0^2 - 4 \cdot 3 \cdot 4 \][/tex]
[tex]\[ D = 0 - 48 \][/tex]
[tex]\[ D = -48 \][/tex]
Now, let's interpret the value of the discriminant:
- If [tex]\( D > 0 \)[/tex], the equation has two distinct real roots.
- If [tex]\( D = 0 \)[/tex], the equation has exactly one real root (a repeated root).
- If [tex]\( D < 0 \)[/tex], the equation has two complex roots.
In this case, since [tex]\( D = -48 \)[/tex] (which is less than 0), the quadratic equation [tex]\( 3x^2 + 4 = 0 \)[/tex] has two complex roots.
Therefore, the correct answer is:
[tex]\[ 2 \][/tex]
