Math Problem Statement
Solution
To solve the given problem, we analyze the quadratic polynomial:
We need to determine for which values of the polynomial has two complex roots.
Step 1: Discriminant condition
The roots of a quadratic polynomial are determined by the discriminant (): Here, for :
Thus, the discriminant becomes:
Step 2: Condition for complex roots
For the quadratic to have two complex roots, the discriminant must be negative: Substitute into the inequality:
Step 3: Solve for
Final Answer:
The polynomial has two complex roots if and only if:
Would you like me to explain further or provide additional examples? Let me know!
5 Related Questions:
- What are the conditions for a quadratic polynomial to have two real roots?
- How would the solution change if the polynomial had one real repeated root instead of complex roots?
- What is the geometric interpretation of the discriminant for quadratic equations?
- How does the value of (in the term ) affect the nature of the roots?
- Can you derive the roots of the equation using the quadratic formula?
Tip:
Always analyze the discriminant () first when solving problems related to the nature of roots for quadratic equations.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Complex Numbers
Formulas
Discriminant formula: Δ = b² - 4ac
Quadratic equation: ax² + bx + c = 0
Theorems
Nature of Roots Theorem
Suitable Grade Level
Grades 9-12
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