The A.M of two numbers is 17 and their G.M. is 8. Find the numbers.

Options

A. 2, 32
B. 4, 30
C. 10, 24
D. None of these

Explanation

To solve the problem, we need to find two numbers \( a \) and \( b \) given that their Arithmetic Mean (A.M.) is 17 and their Geometric Mean (G.M.) is 8.

### Step 1: Set up the equations
The formulas for A.M. and G.M. of two numbers \( a \) and \( b \) are:
- A.M. = \( \frac{a + b}{2} \)
- G.M. = \( \sqrt{ab} \)

From the problem, we know:
1. \( \frac{a + b}{2} = 17 \)
2. \( \sqrt{ab} = 8 \)

### Step 2: Solve for \( a + b \)
From the first equation, we can multiply both sides by 2 to eliminate the fraction:
\[ a + b = 34 \]

### Step 3: Solve for \( ab \)
From the second equation, we square both sides to eliminate the square root:
\[ ab = 8^2 = 64 \]

### Step 4: Form a quadratic equation
Now we have two equations:
1. \( a + b = 34 \)
2. \( ab = 64 \)

We can use these to form a quadratic equation. The general form of a quadratic equation based on the sums and products of the roots is:
\[ x^2 - (a + b)x + ab = 0 \]
Substituting our values:
\[ x^2 - 34x + 64 = 0 \]

### Step 5: Solve the quadratic equation
To find the roots of the quadratic equation, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our case, \( a = 1 \), \( b = -34 \), and \( c = 64 \):
\[ x = \frac{34 \pm \sqrt{(-34)^2 - 4 \cdot 1 \cdot 64}}{2 \cdot 1} \]
Calculating the discriminant:
\[ (-34)^2 = 1156 \]
\[ 4 \cdot 1 \cdot 64 = 256 \]
\[ \sqrt{1156 - 256} = \sqrt{900} = 30 \]
Now substituting back into the quadratic formula:
\[ x = \frac{34 \pm 30}{2} \]

### Step 6: Calculate the two possible values for \( x \)
Calculating the two possible values:
1. \( x = \frac{34 + 30}{2} = \frac{64}{2} = 32 \)
2. \( x = \frac{34 - 30}{2} = \frac{4}{2} = 2 \)

### Step 7: Identify the numbers
Thus, the two numbers are:
- \( a = 32 \) and \( b = 2 \)
- or \( a = 2 \) and \( b = 32 \)

### Final Answer
The two numbers are 32 and 2.

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