To solve for the number of real solutions to the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], you can use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \][/tex]
The part of the formula under the square root, [tex]\( b^2 - 4ac \)[/tex], is called the discriminant. The discriminant lets us determine the number of real solutions to the quadratic equation:
- If the discriminant is positive, there are two distinct real number solutions.
- If the discriminant is zero, there is exactly one real number solution.
- If the discriminant is negative, there are no real number solutions.
In this case, Quincy found the solution to be:
[tex]\[ x = \frac{-3 \pm \sqrt{-19}}{2}. \][/tex]
The discriminant here is the value under the square root:
[tex]\[ -19. \][/tex]
Since the discriminant is negative, it indicates that the quadratic equation has no real number solutions. Negative discriminants result in complex or imaginary solutions, rather than real number solutions.
Therefore, the best description is:
Zero, because the discriminant is negative.
