Complex Numbers And Quadratic Equations

Step 1: Discriminant: D=(2√(2))²-4=8-4=4>0
Step 2: Roots are real and unequal.

Step 1: Substitute y = (x)/(x+1). Transform the equation accordingly.
Step 2: Root relation. New roots become: (α)/(1-α), (β)/(1-β)

Step 1: Use properties of roots of unity.
Since the roots of are cube roots of unity, we know that and will satisfy the same equation as and .
Step 2: Conclusion.
The equation whose roots are and is . Final Answer:

Step 1: For quadratic , roots have equal modulus iff .
Step 2:
Step 3: Hence none of the given roots satisfies the condition.

Step 1: The series is a G.P. with
Step 2:
Step 3: Using

(α-β)²=(α+β)²-4αβ 10=p²-3p p²-3p-10=0 ⟹ p=5,-2

Solving the quadratic equation and comparing moduli of the roots, the root with greater modulus is (5-3i)/(2).

Step 1: For a quadratic x²+px+(3p)/(4)=0: α+β=-p, αβ=(3p)/(4) Step 2: Use: (α-β)²=(α+β)²-4αβ (α-β)²=p²-3p Step 3: Given |α-β|=√(10): p²-3p=10 ⟹ p²-3p-10=0 Step 4: Solving: (p-5)(p+2)=0 ⟹ p=5,-2 ⟹ p∈-2,5

Step 1: Understanding the Question:
The question asks for the absolute value (modulus) of a complex expression involving three complex numbers with given moduli and a specific linear combination modulus.

Step 2: Key Formula or Approach:
For any complex number , we have .
This implies and .
Also, the modulus of a conjugate is the same as the modulus of the number: .

Step 3: Detailed Explanation:
Given .
Given .
Given .
We need to find the modulus of .
Let's factor out :
Using the conjugate relations from
Step 2:
, , .
Substitute these into the expression for S:
Taking the modulus on both sides:
We know .
Substituting the given values:


Step 4: Final Answer:
The absolute value equals 96.

Step 1: Understanding the Concept:
The problem utilizes properties of complex numbers, specifically the relationship . We can factor out the product of the three complex numbers from the expression we need to evaluate.

Step 2: Key Formula or Approach:
Use . For the given magnitudes:

Step 3: Detailed Explanation:
Let . Factor out : We can rewrite the constants using the magnitudes: Taking the absolute value: Since , we have .

Step 4: Final Answer
The absolute value of the expression is 96.