Question

Let two numbers have arithmetic mean $9$ and geometric mean $4$. Then these numbers are the roots of the quadratic equation.

• A. $x^2+18x+16=0$
• B. $x^2-18x+16=0$
• C. $x^2+18x-16=0$
• D. $x^2-18x-16=0$

Answer 1

The correct option is B $x^2-18x+16=0$

Let $\alpha$ and $\beta$ be two numbers whose arithmetic mean is $9$ and the geometric mean is $4$.

$\therefore \alpha +\beta =18$

and $\alpha \beta =16$

$(\because AM\Rightarrow \frac{a+b}{2}andGM\Rightarrow ab=c^2)$

Therefore, the required equation is

$\Rightarrow x^2-(\alpha +\beta )x+\left(\alpha \beta \right)=0$

$\Rightarrow x^2-18x+16=0$