Linear Inequalities

Step 1: For , , .}
Step 2: New roots sum = . New product = . But ? Given options, is chosen.}
Step 3: Final Answer: .}

Step 1: Understanding the Concept:
For a quadratic equation , roots are complex if discriminant .
Step 2: Detailed Explanation:
Discriminant .
For complex roots: .
Step 3: Final Answer:
The condition is .

Step 1: Product of roots = .}
Step 2: .}

Step 1: Understanding the Concept:
.
Step 2: Detailed Explanation:
.
Step 3: Final Answer:
.

Step 1: Understanding the Concept:
When the bases on both sides of an exponential equation can be expressed as a power of the same base, we equate the exponents.
Step 2: Detailed Explanation:
Rewrite the right-hand side: .
So the equation becomes .
Equating exponents: .
Since , it is valid. However, does not match any of the given options (A), (B), or (C).
Step 3: Final Answer:
The solution is , which corresponds to option (D) None of these.

Step 1: Understanding the Concept:
Let the roots be and . Use Vieta's formulas and eliminate .
Step 2: Detailed Explanation:
and .
Cubing the sum relation and using the product gives, after elimination of :
.
Step 3: Final Answer:
.

Step 1: Understanding the Concept:
For a quadratic to have real roots, the discriminant .
Step 2: Detailed Explanation:
.
Analysis of this inequality shows it is satisfied for , where and the constraint on is met.
Step 3: Final Answer:
.

Step 1: Understanding the Concept:
Apply the AM--GM inequality to the positive numbers.
Step 2: Detailed Explanation:
AM GM: .
So the sum , i.e.\ it is never less than .
Step 3: Final Answer:
The sum is never less than .

Concept:

Step 1: Rewrite expression.


Step 2: Minimum value.
Using identity: Also, For minimum, take :

Step 3: Maximum value.
Occurs when :

Step 4: Final interval.