What is the factored form of x2 − 4x − 5?
(x + 5)(x − 1)
(x + 5)(x + 1)
(x − 5)(x − 1)
(x − 5)(x + 1)
The Correct Answer and Explanation is:
To find the factored form of the quadratic expression (x^2 – 4x – 5), we need to factor it into two binomials. The standard approach involves finding two numbers that multiply to give the constant term (-5) and add to give the coefficient of the linear term (-4).
Step 1: Identify the necessary factors
The quadratic expression is in the form:
[
x^2 + bx + c
]
where (b = -4) and (c = -5). To factor the expression, we need to find two numbers that:
- Multiply to give (c = -5)
- Add to give (b = -4)
Step 2: Find the pair of numbers
We need to identify two numbers whose product is (-5) and whose sum is (-4). The factor pairs of (-5) are:
- (1 \times (-5))
- (-1 \times 5)
We now check which of these pairs add up to (-4):
- (1 + (-5) = -4)
- (-1 + 5 = 4)
The pair (1) and (-5) works because:
[
1 + (-5) = -4 \quad \text{and} \quad 1 \times (-5) = -5
]
Step 3: Write the factored form
Since (1) and (-5) are the numbers that satisfy both conditions, we can factor the quadratic as:
[
x^2 – 4x – 5 = (x + 1)(x – 5)
]
Thus, the correct factored form is:
[
(x + 1)(x – 5)
]
Conclusion
The factored form of (x^2 – 4x – 5) is ((x + 1)(x – 5)), which corresponds to option (x − 5)(x + 1). Factoring quadratics involves finding two numbers that multiply to the constant term and add to the coefficient of the linear term.