To determine the nature of the solutions of the quadratic equation [tex]\(3x^2 + 5x + 7 = 0\)[/tex], we use the discriminant.
The discriminant [tex]\(D\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by the formula:
[tex]\[ D = b^2 - 4ac \][/tex]
In our equation, the coefficients are:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 5 \)[/tex]
- [tex]\( c = 7 \)[/tex]
Substituting these into the formula for the discriminant:
[tex]\[ D = (5)^2 - 4(3)(7) \][/tex]
Now, compute each part step-by-step:
[tex]\[ 5^2 = 25 \][/tex]
[tex]\[ 4 \cdot 3 \cdot 7 = 84 \][/tex]
So, the discriminant is:
[tex]\[ D = 25 - 84 = -59 \][/tex]
The value of the discriminant [tex]\(D\)[/tex] tells us about the nature of the roots of the quadratic equation:
- If [tex]\(D > 0\)[/tex], there are two distinct real solutions.
- If [tex]\(D = 0\)[/tex], there is one real solution (a repeated root).
- If [tex]\(D < 0\)[/tex], there are no real solutions, but there are two complex solutions.
Since in our case [tex]\(D = -59\)[/tex], which is less than 0, the quadratic equation [tex]\(3x^2 + 5x + 7 = 0\)[/tex] has no real solutions, but it has two complex solutions.
Thus, the correct choice is:
- 2 complex solutions
