Question

form a quadratic equation in X such that the arithmetic mean of its roots is m and their geometric mean is n ​

Answer 1

ANSWER

Let the two roots be α and β

∴

2

α+β

=8 (AM of a and b is

2

a+b

) ⟶(1)

and

αβ

=5 (GM of a and b is

ab

) ⟶(2)

From (1)

α+β=16

⇒ sum of roots =α+β=16

From (2)

αβ

=5

⇒αβ=25

∴ Products of roots =25

Now, we know that if we have the sum of roots and product of roots then the quadratic equation is given by

x

2

− (sum of the roots)x + (product of the roots) =0

∴ the quadratic equation with roots α,β are

x

2

−(α+β)x+αβ=0

x

2

−16x+25=0

Answer 2

Answer:

your answer

Step-by-step explanation:

Let α and β are the roots of the equation

Given arithmetic mean of its roots is m

then, (α + β)/2 = m

∴ α + β = 2m eq. (1)

geometric mean of the roots is n

√(αβ) = n

∴ αβ = n² eq.(2)

we know that quadratic equation is of the form

x^{2} + (α + β)x + αβ = 0

From eq 1 and eq 2 x^{2} + 2mx + n² = 0