Find two consecutive odd natural numbers , the sum of whose squares is 290.

Answer

To find two consecutive odd natural numbers whose squares sum up to 290, we can follow these steps:

### Step 1: Define the Numbers
Let the first odd natural number be \( x \). Since we are looking for consecutive odd numbers, the next consecutive odd number will be \( x + 2 \).

### Step 2: Set Up the Equation
According to the problem, the sum of the squares of these two numbers is 290. Therefore, we can write the equation:
\[ x^2 + (x + 2)^2 = 290 \]

### Step 3: Expand the Equation
Now, we will expand the equation:
\[ x^2 + (x^2 + 4x + 4) = 290 \]
This simplifies to:
\[ 2x^2 + 4x + 4 = 290 \]

### Step 4: Rearrange the Equation
Next, we will rearrange the equation to set it to zero:
\[ 2x^2 + 4x + 4 - 290 = 0 \]
This simplifies to:
\[ 2x^2 + 4x - 286 = 0 \]

### Step 5: Simplify the Equation
We can divide the entire equation by 2 to make it simpler:
\[ x^2 + 2x - 143 = 0 \]

### Step 6: Factor the Quadratic Equation
Now, we will factor the quadratic equation. We need to find two numbers that multiply to -143 and add to 2. The numbers are 13 and -11.
Thus, we can factor it as:
\[ (x + 13)(x - 11) = 0 \]

### Step 7: Solve for \( x \)
Setting each factor to zero gives us:
\[ x + 13 = 0 \quad \text{or} \quad x - 11 = 0 \]
So, we find:
\[ x = -13 \quad \text{or} \quad x = 11 \]

### Step 8: Determine the Odd Natural Numbers
Since we are looking for natural numbers, we discard \( x = -13 \) and take \( x = 11 \). Therefore, the two consecutive odd natural numbers are:
\[ 11 \quad \text{and} \quad 13 \]

### Step 9: Verify the Solution
Finally, we verify our solution by checking the sum of their squares:
\[ 11^2 + 13^2 = 121 + 169 = 290 \]
This confirms that our solution is correct.

### Final Answer
The two consecutive odd natural numbers are **11 and 13**.

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