Math Problem Statement

For which values of c does the following polynomial have two complex roots?

Solution

To solve the given problem, we analyze the quadratic polynomial:

x2+4x+cx^2 + 4x + c

We need to determine for which values of cc the polynomial has two complex roots.

Step 1: Discriminant condition

The roots of a quadratic polynomial are determined by the discriminant (Δ\Delta): Δ=b24ac\Delta = b^2 - 4ac Here, for x2+4x+cx^2 + 4x + c:

  • a=1a = 1
  • b=4b = 4
  • c=cc = c

Thus, the discriminant becomes: Δ=424(1)(c)=164c\Delta = 4^2 - 4(1)(c) = 16 - 4c

Step 2: Condition for complex roots

For the quadratic to have two complex roots, the discriminant must be negative: Δ<0\Delta < 0 Substitute Δ=164c\Delta = 16 - 4c into the inequality: 164c<016 - 4c < 0

Step 3: Solve for cc

16<4c    c>416 < 4c \quad \implies \quad c > 4

Final Answer:

The polynomial x2+4x+cx^2 + 4x + c has two complex roots if and only if: c>4c > 4


Would you like me to explain further or provide additional examples? Let me know!


5 Related Questions:

  1. What are the conditions for a quadratic polynomial to have two real roots?
  2. How would the solution change if the polynomial had one real repeated root instead of complex roots?
  3. What is the geometric interpretation of the discriminant for quadratic equations?
  4. How does the value of bb (in the term bxbx) affect the nature of the roots?
  5. Can you derive the roots of the equation x2+4x+cx^2 + 4x + c using the quadratic formula?

Tip:

Always analyze the discriminant (Δ=b24ac\Delta = b^2 - 4ac) 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