Question

Two numbers \( x \) and \( y \) have arithmetic mean 9 and geometric mean 4. Then \( x \) and \( y \) are the roots of:

• A. \( x^2 - 18x - 16 = 0 \)
• B. \( x^2 - 18x + 16 = 0 \)
• C. \( x^2 + 18x - 16 = 0 \)
• D. \( x^2 + 18x + 16 = 0 \)
• E. \( x^2 - 17x + 16 = 0 \)

Hint:

The relation \( x^2 - 2(\text{A.M.})x + (\text{G.M.})^2 = 0 \) is a direct shortcut to finding the equation when means are given. It skips the intermediate step of finding the sum and product separately.

Answer 1

The Correct Option is B

Concept: A quadratic equation can be formed if the sum and product of its roots are known. The standard form is \( x^2 - (\text{Sum of Roots})x + (\text{Product of Roots}) = 0 \). The arithmetic mean (A.M.) and geometric mean (G.M.) provide these values directly.

Step 1: Determining the sum and product of the numbers.
Given the A.M. of \( x \) and \( y \) is 9: \[ \frac{x + y}{2} = 9 \implies x + y = 18 \] Given the G.M. of \( x \) and \( y \) is 4: \[ \sqrt{xy} = 4 \implies xy = 16 \] The sum of roots is 18 and the product of roots is 16.

Step 2: Formulating the quadratic equation.
Substitute the sum and product into the quadratic template: \[ x^2 - (18)x + (16) = 0 \] \[ x^2 - 18x + 16 = 0 \]