Question

If the arithmetic mean of the roots of a quadratic equation is $8$ and the geometric mean is $5$, then the equation is

• A. $x^2-16x-25=0$
• B. $x^2+16x-25=0$
• C. $x^2-16x+25=0$
• D. $x^2-8x-5=0$

Answer 1

The correct option is C

$x^2-16x+25=0$

Explanation for the correct option.

Step 1: Find the sum and product of roots.

Given that, the arithmetic mean of the roots of a quadratic equation is $8$ and the geometric mean is $5$.

Let the roots of the quadratic equation be $\alpha ,\beta$.

By definition of arithmetic & geometric mean we have:

$$\begin{gathered} \frac{\alpha +\beta }{2}=8 \\ \Rightarrow \alpha +\beta =16 \end{gathered}$$

And $\begin{array}{rcl}\sqrt{\alpha \beta }& =& 5\end{array}$

$\begin{array}{rcl}& \Rightarrow & \alpha \beta =25\end{array}$

Step 2: Find the quadratic equation.

We know that the quadratic equation is of the form:

$x^2-\left(\mathrm{sum}\mathrm{of}\mathrm{roots}\right)\mathrm{x}+\left(\mathrm{product}\mathrm{of}\mathrm{roots}\right)=0$

On substituting the value of sum and product of root from step in general form of quadratic equation, we get;

${\mathrm{x}}^2-16\mathrm{x}+25=0$

Hence, option C is correct.