Factorise : ` x^(2) + 5 x - 14`

Options

A. `( x - 7) ( x + 2)`
B. ` ( x - 1) ( x + 14)`
C. ` ( x + 7) ( x - 2)`
D. ` ( x - 14) ( x + 1)`

Answer

To factorize the quadratic expression \( x^2 + 5x - 14 \), we will follow these steps:

### Step 1: Identify the coefficients
The given expression is in the form \( ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = 5 \)
- \( c = -14 \)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that:
- Multiply to \( ac = 1 \times (-14) = -14 \)
- Add up to \( b = 5 \)

The two numbers that satisfy these conditions are \( 7 \) and \( -2 \):
- \( 7 \times (-2) = -14 \)
- \( 7 + (-2) = 5 \)

### Step 3: Rewrite the middle term using the two numbers
We can rewrite the expression \( x^2 + 5x - 14 \) by splitting the middle term:
\[ x^2 + 7x - 2x - 14 \]

### Step 4: Group the terms
Now, we group the terms:
\[ (x^2 + 7x) + (-2x - 14) \]

### Step 5: Factor by grouping
Now we factor out the common factors from each group:
- From the first group \( (x^2 + 7x) \), we can factor out \( x \):
\[ x(x + 7) \]
- From the second group \( (-2x - 14) \), we can factor out \( -2 \):
\[ -2(x + 7) \]

Now, we can combine these:
\[ x(x + 7) - 2(x + 7) \]

### Step 6: Factor out the common binomial
Now, we can factor out the common binomial \( (x + 7) \):
\[ (x + 7)(x - 2) \]

### Final Result
Thus, the factorization of \( x^2 + 5x - 14 \) is:
\[ (x + 7)(x - 2) \]