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Question
If AM and GM of two positive numbers a and b are 10 and 8 respectively, find the numbers.
Solution
\[AM = 10\]
\[ \therefore \frac{a + b}{2} = 10\]
\[ \Rightarrow a + b = 20 . . . . . . . . (i)\]
\[\text { Also }, G = 8\]
\[ \therefore \sqrt{ab} = 8\]
\[ \Rightarrow ab = 8^2 \]
\[ \Rightarrow ab = 64 . . . . . . . . (ii)\]
\[\text { Using (i) and (ii) }: \]
\[ \Rightarrow a\left( 20 - a \right) = 64\]
\[ \Rightarrow a^2 - 20a + 64 = 0\]
\[ \Rightarrow a^2 - 16a - 4a + 64 = 0\]
\[ \Rightarrow a\left( a - 16 \right) - 4\left( a - 16 \right) = 0\]
\[ \Rightarrow \left( a - 16 \right)\left( a - 4 \right) = 0\]
\[ \Rightarrow a = 4, 16\]
\[\text { If a = 4, then b = 16 } . \]
\[\text { And, if a = 16, then b = 4 .} \]
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