Question

9) If arithmetic mean of two numbers a and bis 8 and ab = 9, find a quadratic equation whose
roots are a and be
=X2- 8x + 9 = 0
= x2 – 16x + 9 = 0
Ο Ο Ο
= x2 - 4x + 9 = 0
O = x2 +16x + 9 = 0​

Answer 1

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Asked on December 20, 2019 by

Sonu Shelke

If the Arithmetic mean and Geometric mean of roots of a quadratic equation are 8 and 5 respectively then obtain the quadratic equation.

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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