Question

Arithmetic and Geometric mean of roots of a Quadratic Equation are 8 and 5. Find the equation.​

Answer 1

Let the roots of the quadratic equation be aa and bb

It is given that the A.M of the roots =8=8 and their G.M.=5=5

We know that the A.M between aa and b=a+b2b=a+b2 and

G.M.=ab−−√=ab

⇒a+b2⇒a+b2=8=8 and ab−−√=5ab=5

⇒a+b=16⇒a+b=16 and ab=25ab=25

Since aa and bb are roots of the quadratic equation,

sum of the roots =a+b=16=a+b=16

Product of the roots =ab=25

Answer 2

$\huge \boxed{ \underline{ \underline{ \bf{Answer}}}}$

Arithmetic mean of roots of Quadratic Equation = 8.

Geometric mean of roots of Quadratic Equation = 5.

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

A/Q,

For Arithmetic mean,

$\implies \: \frac{ \alpha \: + \: \beta }{2} = \: 8$

$\implies \: \alpha \: + \: \beta = \: 16 \: \: \:$

For Geometric mean,

$\implies \: \: \sqrt{ \alpha \beta } \: = \: 5$

Squaring, both sides

$\implies \: { (\sqrt{ \alpha \beta }) }^{2} = {5}^{2}$

$\implies \: \alpha \beta \: = \: 25 \: \: \: \: \: \: \:$

We know, for Quadratic Equations

Sum of the zeroes($\alpha + \beta$) = -b/a and

Product of zeroes ($\alpha \times \beta$) = c/a.

Hence, our required Quadratic Equation is

$\implies \: {x}^{2} - ( \alpha + \beta )x + \alpha \beta = 0$

$\boxed{ \sf{ \implies \: {x}^{2} - 16x + 25 = 0}}$