Question

What will be the quadratic equation x when the roots have arithmetic mean A and geometric mean B

• A. x²+ax+b²=0
• B. x²-ax+b=0
• C. x²_2ax+b²=0
• D. x²+ax+b=0

Answer 1

The Correct Option is C

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:
$x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$
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:
$\frac{α + β}{2} = A \implies α + β = 2A$
Given their geometric mean:
$\sqrt{αβ} = B \implies αβ = B^2$

3. Forming the Quadratic Equation:
Using the standard form of a quadratic equation with roots α and β:
$x^2 - (α + β)x + αβ = 0$
Substituting the values we found:
$x^2 - 2A x + B^2 = 0$

4. Verification:
For any quadratic equation $x^2 - px + q = 0$:
- Sum of roots = p = 2A (matches our expression)
- Product of roots = q = B² (matches our expression)

Final Answer:
The required quadratic equation is $\boxed{x^2 - 2A x + B^2 = 0}$.