If A.M. and G.M. of two positive numbers $a$ and $b$ are 10 and 8, respectively, find the numbers.

Step 1: Formulate equations from given information

The problem states that the arithmetic mean (A.M.) of two positive numbers $a$ and $b$ is 10, and their geometric mean (G.M.) is 8. We can write these as two equations. The formula for A.M. is $(a+b)/2$ and for G.M. is $\sqrt{ab}$.

Step 2: Simplify the equations

From equation (1), we multiply both sides by 2 to get $a+b=20$. From equation (2), we square both sides to eliminate the square root, resulting in $ab=64$. These simplified equations will be easier to work with.

Step 3: Express one variable in terms of the other

From equation (3), we can express $b$ in terms of $a$. This allows us to substitute $b$ into the other equation, reducing the system to a single variable.

Step 4: Substitute and form a quadratic equation

Substitute the expression for $b$ from equation (5) into equation (4). This will give us a quadratic equation in terms of $a$. Rearranging the terms, we get $a^2 - 20a + 64 = 0$.

Step 5: Solve the quadratic equation for 'a'

We can solve this quadratic equation by factoring. We need two numbers that multiply to 64 and add up to -20. These numbers are -4 and -16. So, the equation factors into $(a-4)(a-16)=0$, giving us two possible values for $a$: 4 or 16.

Step 6: Find the corresponding values for 'b'

Now, we use equation (5), $b = 20 - a$, to find the corresponding values for $b$. If $a=4$, then $b=16$. If $a=16$, then $b=4$. Both pairs satisfy the original conditions.

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