To find the possible expressions for the length and breadth of a rectangular field with an area of \( 14x^2 - 11x - 15 \, m^2 \), we need to factor the quadratic expression. Here’s a step-by-step solution:
### Step 1: Write down the quadratic expression
We start with the area of the rectangular field given by:
\[
A = 14x^2 - 11x - 15
\]
### Step 2: Identify the coefficients
In the quadratic expression \( ax^2 + bx + c \):
- \( a = 14 \)
- \( b = -11 \)
- \( c = -15 \)
### Step 3: Multiply \( a \) and \( c \)
We need to find two numbers that multiply to \( a \times c = 14 \times (-15) = -210 \) and add up to \( b = -11 \).
### Step 4: Find the factors
We look for pairs of factors of \(-210\) that add up to \(-11\):
- The pairs of factors of \(-210\) are:
- \( (1, -210) \)
- \( (-1, 210) \)
- \( (2, -105) \)
- \( (-2, 105) \)
- \( (3, -70) \)
- \( (-3, 70) \)
- \( (5, -42) \)
- \( (-5, 42) \)
- \( (6, -35) \)
- \( (-6, 35) \)
- \( (7, -30) \)
- \( (-7, 30) \)
- \( (10, -21) \)
- \( (-10, 21) \)
- \( (14, -15) \)
- \( (-14, 15) \)
From these pairs, we find that \( 10 \) and \( -21 \) add up to \( -11 \).
### Step 5: Rewrite the middle term
We can rewrite the expression \( 14x^2 - 11x - 15 \) as:
\[
14x^2 + 10x - 21x - 15
\]
### Step 6: Factor by grouping
Now, we group the terms:
\[
(14x^2 + 10x) + (-21x - 15)
\]
Factoring out the common factors in each group:
\[
2x(7x + 5) - 3(7x + 5)
\]
Now we can factor out \( (7x + 5) \):
\[
(7x + 5)(2x - 3)
\]
### Step 7: Identify length and breadth
Thus, the area \( 14x^2 - 11x - 15 \) can be expressed as:
\[
(7x + 5)(2x - 3)
\]
Here, we can consider:
- Length = \( 7x + 5 \)
- Breadth = \( 2x - 3 \)
### Conclusion
The possible expressions for the length and breadth of the field are:
- Length: \( 7x + 5 \)
- Breadth: \( 2x - 3 \)
