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

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Solution

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

1 $x^{2} + 18 x - 16 = 0$

2 $x^{2} - 18 x + 16 = 0$

3 $x^{2} + 18 x + 16 = 0$

4 $x^{2} - 18 x - 16 = 0$

Ans.
(2)

$Let the two numbers be α, β .$

$∴$ $\frac{α + β}{2} = 9$ $and$ $\sqrt{α β} = 4$

$∴$ $Required equation is$

$x^{2} - 2 (Average value of α, β) x + {(\sqrt{G.M.})}^{2} = 0$

$∴$ $x^{2} - 2 (9) x + 16 = 0$ $\Rightarrow x^{2} - 18 x + 16 = 0$

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