If arithmetic and geometric mean of two positive numbers are 5 and 3, then numbers are:

Options

A. 9, 1
B. 4, 16
C. 4, 8
D. 2, 4

Answer

To solve the problem, we need to find two positive numbers \( a \) and \( b \) given that their arithmetic mean (AM) is 5 and their geometric mean (GM) is 3.

### Step 1: Set up the equations
From the problem, we know:
- The arithmetic mean is given by:
\[ \text{AM} = \frac{a + b}{2} = 5 \]
- The geometric mean is given by:
\[ \text{GM} = \sqrt{ab} = 3 \]

### Step 2: Solve for \( a + b \)
From the arithmetic mean equation, we can multiply both sides by 2 to eliminate the fraction:
\[ a + b = 10 \]

### Step 3: Solve for \( ab \)
From the geometric mean equation, we can square both sides to eliminate the square root:
\[ ab = 3^2 = 9 \]

### Step 4: Form a quadratic equation
Now we have two equations:
1. \( a + b = 10 \)
2. \( ab = 9 \)

We can use these to form a quadratic equation. The standard form of a quadratic equation based on the sum and product of roots is:
\[ x^2 - (a + b)x + ab = 0 \]
Substituting our values:
\[ x^2 - 10x + 9 = 0 \]

### Step 5: Solve the quadratic equation
To find the roots of the quadratic equation, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -10 \), and \( c = 9 \):
\[ x = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 9}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{10 \pm \sqrt{100 - 36}}{2} \]
\[ x = \frac{10 \pm \sqrt{64}}{2} \]
\[ x = \frac{10 \pm 8}{2} \]

### Step 6: Find the values of \( a \) and \( b \)
Calculating the two possible values:
1. \( x = \frac{10 + 8}{2} = \frac{18}{2} = 9 \)
2. \( x = \frac{10 - 8}{2} = \frac{2}{2} = 1 \)

Thus, the two numbers are \( a = 9 \) and \( b = 1 \).

### Final Answer
The two positive numbers are \( 9 \) and \( 1 \).

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