Key Concepts and Formulas

• Arithmetic Mean (AM) of two numbers $a$ and $b$: ${{a + b} \over 2}$
• Geometric Mean (GM) of two numbers $a$ and $b$: $\sqrt{ab}$
• Quadratic equation with roots $a$ and $b$: ${x^2} - (a+b)x + ab = 0$

Step-by-Step Solution

Step 1: Find the sum of the roots using the Arithmetic Mean. We are given that the arithmetic mean of the two numbers is 9. We will use the formula for the arithmetic mean to find the sum of the roots. ${{a + b} \over 2} = 9$ Explanation: The arithmetic mean is directly related to the sum of the numbers. Multiplying the AM by 2 will give the sum. Multiplying both sides by 2: $a + b = 18$ So, the sum of the roots is 18.

Step 2: Find the product of the roots using the Geometric Mean. We are given that the geometric mean of the two numbers is 4. We will use the formula for the geometric mean to find the product of the roots. $\sqrt{ab} = 4$ Explanation: The geometric mean is directly related to the product of the numbers. Squaring the GM will give the product. Squaring both sides of the equation: $(\sqrt{ab})^2 = 4^2$ $ab = 16$ So, the product of the roots is 16.

Step 3: Form the quadratic equation. Now that we have the sum of the roots ($a+b=18$) and the product of the roots ($ab=16$), we can substitute these values into the general form of a quadratic equation: ${x^2} - (\text{sum of roots})x + (\text{product of roots}) = 0$ ${x^2} - (18)x + (16) = 0$ ${x^2} - 18x + 16 = 0$ Explanation: This step uses the relationship between the roots and coefficients of a quadratic equation.

Common Mistakes & Tips

• Sign Error: Remember the equation is $x^2 - (\text{sum})x + (\text{product}) = 0$. The coefficient of the x term is the negative of the sum.
• Squaring: Ensure you square the entire geometric mean value to find the product.
• AM-GM: Recall that $AM \ge GM$. Here $9 \ge 4$, which is consistent.

Summary

By utilizing the definitions of arithmetic and geometric means, we found the sum and product of the roots of the desired quadratic equation. Substituting these values into the standard quadratic equation form, we get the equation ${x^2} - 18x + 16 = 0$.

The final answer is \boxed{${x^2} - 18x + 16 = 0$}, which corresponds to option (B).