Factorise `x^(2) + 10x + 25`.

Options

A. `(x+25)^(2)`
B. `(x+5)^(2)`
C. `(x+5) (x+2)`
D. `(x+5) (x+3)`

Answer

To factorise the expression \( x^2 + 10x + 25 \), we can follow these steps:

### Step 1: Identify the components of the quadratic expression
The expression is in the form \( ax^2 + bx + c \), where:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 10 \) (coefficient of \( x \))
- \( c = 25 \) (constant term)

### Step 2: Rewrite the middle term
We can express the middle term \( 10x \) as \( 2 \cdot x \cdot 5 \). This helps us see the relationship between the terms.

### Step 3: Recognize the perfect square
Notice that \( 25 \) is the square of \( 5 \) (i.e., \( 5^2 \)). Therefore, we can rewrite the expression as:
\[ x^2 + 2 \cdot x \cdot 5 + 5^2 \]

### Step 4: Apply the algebraic identity
The expression \( a^2 + 2ab + b^2 \) can be factored using the identity \( (a + b)^2 \). Here, \( a = x \) and \( b = 5 \). Thus, we can factor the expression as:
\[ (x + 5)^2 \]

### Step 5: Write the final factorised form
The factorised form of the expression \( x^2 + 10x + 25 \) is:
\[ (x + 5)(x + 5) \quad \text{or} \quad (x + 5)^2 \]

### Final Answer:
\[ x^2 + 10x + 25 = (x + 5)^2 \]

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