Find 2 consecutive natural numbers, sum of whose squares is 25.

Options

A. `2,3`
B. `3,4`
C. `4,5`
D. `1,4`

Answer

To find two consecutive natural numbers whose sum of squares is 25, we can follow these steps:

### Step 1: Define the consecutive natural numbers
Let the first consecutive natural number be \( x \). Therefore, the next consecutive number will be \( x + 1 \).

### Step 2: Write the equation for the sum of squares
The sum of the squares of these two numbers can be expressed as:
\[ x^2 + (x + 1)^2 = 25 \]

### Step 3: Expand the equation
Now, we will expand the equation:
\[ x^2 + (x^2 + 2x + 1) = 25 \]
This simplifies to:
\[ 2x^2 + 2x + 1 = 25 \]

### Step 4: Rearrange the equation
Next, we will rearrange the equation to set it to zero:
\[ 2x^2 + 2x + 1 - 25 = 0 \]
This simplifies to:
\[ 2x^2 + 2x - 24 = 0 \]

### Step 5: Divide the entire equation by 2
To simplify, we can divide the entire equation by 2:
\[ x^2 + x - 12 = 0 \]

### Step 6: Factor the quadratic equation
Now, we will factor the quadratic equation:
\[ (x + 4)(x - 3) = 0 \]

### Step 7: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x + 4 = 0 \) → \( x = -4 \) (not a natural number)
2. \( x - 3 = 0 \) → \( x = 3 \) (valid solution)

### Step 8: Find the consecutive numbers
Now that we have \( x = 3 \), the consecutive numbers are:
- First number: \( 3 \)
- Second number: \( 3 + 1 = 4 \)

### Conclusion
The two consecutive natural numbers whose sum of squares is 25 are **3 and 4**.

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