Let two numbers have arithmatic mean 9 and geometric mean 4.Then these numbers are roots of the equation (a) `x^2+18x+16=0` (b) `x^2-18x-16=0` (c) `x^2+18x-16=0` (d) `x^2-18x+16=0`

Options

A. `x ^(2) +18x + 16 =0`
B. `x ^(2) -18x - 16 =0`
C. `x ^(2) + 18x -16 =0`
D. `x ^(2) -18x +16=0`

Answer

To solve the problem, we need to find two numbers based on the given arithmetic mean and geometric mean, and then determine the corresponding quadratic equation for those numbers.

### Step 1: Set up the equations
Given:
- Arithmetic Mean (AM) = 9
- Geometric Mean (GM) = 4

From the definition of the arithmetic mean:
\[ \frac{a + b}{2} = 9 \]
Multiplying both sides by 2 gives:
\[ a + b = 18 \quad \text{(1)} \]

From the definition of the geometric mean:
\[ \sqrt{ab} = 4 \]
Squaring both sides gives:
\[ ab = 16 \quad \text{(2)} \]

### Step 2: Form the quadratic equation
We know that if \(a\) and \(b\) are the roots of the quadratic equation, the equation can be expressed as:
\[ x^2 - (a + b)x + ab = 0 \]
Substituting the values from equations (1) and (2):
\[ x^2 - 18x + 16 = 0 \]

### Step 3: Identify the correct option
The equation we derived is:
\[ x^2 - 18x + 16 = 0 \]
Now, let's compare this with the given options:
(a) \(x^2 + 18x + 16 = 0\)
(b) \(x^2 - 18x - 16 = 0\)
(c) \(x^2 + 18x - 16 = 0\)
(d) \(x^2 - 18x + 16 = 0\)

The correct option is (d) \(x^2 - 18x + 16 = 0\).

### Final Answer:
The correct option is (d) \(x^2 - 18x + 16 = 0\).

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