Quadratic Equations

To find the sum and product of the roots for the quadratic equation , we use Vieta's formulas, which relate the coefficients of a quadratic equation to the sum and product of its roots. For any quadratic equation of the form , the sum of the roots ( ) is given by and the product of the roots ( ) is given by .

Given equation:

Here, , , and .

• Sum of the roots:
• Product of the roots:

Thus, the sum and product of the roots are and , respectively. Therefore, the correct option is -0,4.

To solve the problem, we need to find the quadratic equation whose roots have a given arithmetic mean A and geometric mean B.

1. Understanding the Problem:
We need to find a quadratic equation of the form:

where the roots have:
- Arithmetic mean = A
- Geometric mean = B

2. Expressing the Sum and Product of Roots:
Let the roots be α and β.
Given their arithmetic mean:

Given their geometric mean:

3. Forming the Quadratic Equation:
Using the standard form of a quadratic equation with roots α and β:

Substituting the values we found:

4. Verification:
For any quadratic equation :
- Sum of roots = p = 2A (matches our expression)
- Product of roots = q = B² (matches our expression)

Final Answer:
The required quadratic equation is .