What is the value of 2 consecutive natural numbers sum of whose squares is 145 ?

Options

A. `8,9`
B. `6,7`
C. `13,14`
D. `9,10`

Answer

To find the value of two consecutive natural numbers whose sum of squares is 145, we can follow these steps:

### Step 1: Define the consecutive natural numbers
Let the first natural number be \( x \). Therefore, the next consecutive natural 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:
\[ x^2 + (x + 1)^2 = 145 \]

### Step 3: Expand the equation
Now, we will expand the equation:
\[ x^2 + (x^2 + 2x + 1) = 145 \]
This simplifies to:
\[ 2x^2 + 2x + 1 = 145 \]

### Step 4: Rearrange the equation
Next, we will rearrange the equation to set it to zero:
\[ 2x^2 + 2x + 1 - 145 = 0 \]
This simplifies to:
\[ 2x^2 + 2x - 144 = 0 \]

### Step 5: Divide the entire equation by 2
To simplify the equation, we can divide everything by 2:
\[ x^2 + x - 72 = 0 \]

### Step 6: Factor the quadratic equation
Now, we will factor the quadratic equation:
\[ (x + 9)(x - 8) = 0 \]

### Step 7: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x + 9 = 0 \) → \( x = -9 \) (not a natural number)
2. \( x - 8 = 0 \) → \( x = 8 \)

### Step 8: Identify the consecutive numbers
Since \( x = 8 \), the two consecutive natural numbers are:
- First number: \( 8 \)
- Second number: \( 8 + 1 = 9 \)

### Final Answer
The two consecutive natural numbers are **8 and 9**.

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