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Question
Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively.
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Solution
\[\text{ Let the roots of the required quadratic equation be a and b } . \]
\[ \therefore A = \frac{a + b}{2} and G = \sqrt{ab}\]
\[\text{ The equation having a and b as its roots is }\]
\[ x^2 - x\left( a + b \right) + ab = 0\]
\[ \Rightarrow x^2 - 2Ax + G^2 = 0 \left[ \because A = \frac{a + b}{2} \text{ and } G = \sqrt{ab} \right]\]
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