If the discriminant of a quadratic equation is equal to -8, which statement describes the roots?
A. There are two complex roots.
B. There are two real roots.
C. There is one real root.
D. There is one complex root.
Answer (2)
The discriminant is used to determine the nature of the roots of a quadratic equation. Since the discriminant is -8, which is negative, the quadratic equation has two complex roots. Therefore, the correct answer is that there are two complex roots.
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The discriminant of a quadratic equation determines the nature of its roots.
A negative discriminant indicates complex roots.
Since the discriminant is -8 (negative), the quadratic equation has two complex roots.
Therefore, the answer is: There are two complex roots. T h ere a re tw oco m pl e x roo t s .
Explanation
Understanding the Discriminant The discriminant of a quadratic equation determines the nature of its roots. The discriminant, denoted as Δ , is given by the expression b 2 − 4 a c in the quadratic formula x = 2 a − b ± b 2 − 4 a c .
Relating the Discriminant to the Roots If the discriminant 0"> Δ > 0 , the quadratic equation has two distinct real roots. If Δ = 0 , the quadratic equation has one real root (a repeated root). If Δ < 0 , the quadratic equation has two complex conjugate roots.
Determining the Nature of the Roots In this problem, the discriminant is given as -8, which is a negative number. Therefore, according to the rules, the quadratic equation has two complex roots.
Examples
Understanding the discriminant is crucial in various fields, such as physics and engineering, where quadratic equations are used to model physical phenomena. For example, in projectile motion, the discriminant can determine whether a projectile will hit a target or not. If the discriminant is negative, it means the projectile will not reach the target, indicating complex roots that have no physical meaning in this context. This helps engineers design systems that meet specific requirements.
