Quadratic Equations

Given, :


Roots are irrational, real and unequal.
So, the correct option is (C) : real, irrational and unequal.

We are given the quadratic equation: where and are odd integers, and . To find the nature of the roots, we use the discriminant of the quadratic equation, which is given by: For the quadratic equation , we have: Substitute these values into the discriminant formula: Simplifying the terms: Now expand the square: This simplifies to:

Step 1: Analyzing the discriminant
Since and are odd integers, the expression will always be a perfect square. Specifically, it will be a non-negative integer, which means the discriminant is a perfect square.

Step 2: Roots of the quadratic equation
For a quadratic equation, the roots are real if , and they are non-real if . Since is a perfect square, the roots will be real, but not necessarily equal.

Conclusion:
The roots of the quadratic equation are real, and since the discriminant is a perfect square, they are distinct and real. Therefore, the roots are said to be rotational in the sense that they correspond to a real rotation in the complex plane.

Answer:

We are given the quadratic equation: Let the roots of the equation be and . We need to find the value of for which the sum of the squares of the roots is minimized.

Step 1: Sum and product of the roots
From Vieta's formulas for a quadratic equation , we know that: For the equation , comparing with , we get:

Step 2: Expression for the sum of the squares of the roots
The sum of the squares of the roots is given by: Substituting the values from Vieta's formulas: Expanding this expression:

Step 3: Minimizing the sum of squares of the roots
The function to minimize is: To find the value of that minimizes this function, we take the derivative of : Setting to find the critical points:

Step 4: Verifying the minimum
To confirm that gives a minimum, we check the second derivative: Since , the function is concave up, and corresponds to a minimum.

Answer:

Step 1: Given a quadratic expression: has no real roots

Step 2: For no real roots, the discriminant must be less than 0:


Step 3: Let us check which sequence among A.P., G.P., and H.P. satisfies this condition when , , are in that sequence.

Case 1: A.P.
If , , are in Arithmetic Progression, then:

Substitute into discriminant:

This value is not necessarily negative for all , .

Case 2: G.P.
If , , are in Geometric Progression, then:

Substitute into discriminant:

This is negative if , .

However, the question specifies the condition , i.e.,
— this contradicts the requirement that

Case 3: H.P.
If , , are in Harmonic Progression, then:

Substitute into discriminant:
, and we test whether:

We simplify:

Take common denominator:


Since for , the numerator is negative, so


Conclusion: The condition is only satisfied when , , are in H.P.

Final Answer: (C): , , cannot be in A.P. or G.P., but can be in H.P.

1. The given quadratic equation is: .
2. Sum of roots = , and Product of roots = (by Vieta’s formulas).
3. We’re told that one root is less than 2 and the other is greater than 2.
4. This means the roots are on either side of 2, so the function changes sign at .
5. Let the roots be and , such that .
6. Then their product implies the roots have opposite signs.
7. Since , this is valid.
8. Also, the sum (since both are around 2), so → p is negative.
9. Now, the expression to evaluate is: .
10. Since and , this expression is less than zero.

Therefore, the correct condition is:

The correct option is (D)

We are given the quadratic equation:

Step 1: General Form of the Quadratic Equation

The given quadratic equation is in the form:

where and .

Step 2: Use of Vieta's Formulas

For the equation , Vieta’s formulas give the relationships between the coefficients and the roots. Let the roots be and . According to Vieta’s formulas:

and

In our case:

and

Step 3: Sum of the Squares of the Roots

We are asked to find the sum of the squares of the roots, which is given by:

Using the identity:

Substitute the values from Vieta's formulas:

Simplifying:

Expanding:

Step 4: Minimize the Expression

Now, we need to minimize the expression . This is a quadratic expression, and we know that a quadratic function of the form is minimized at .

For our expression , we have , , and . The value of that minimizes the expression is:

Step 5: Conclusion

The value of that minimizes the sum of the squares of the roots is .