Factor $x^2-2 x-80$
A) $(x+6)(x-1)$
B) $(x+8)(x-10)$
C) $(x-8)(x+10)$
D) $(x+3)(x+6)$
Answer (1)
We need to factor the quadratic expression x 2 − 2 x − 80 .
We look for two numbers that multiply to -80 and add to -2. These numbers are 8 and -10.
Therefore, the factored form is ( x + 8 ) ( x − 10 ) .
The final answer is ( x + 8 ) ( x − 10 ) .
Explanation
Understanding the Problem We are asked to factor the quadratic expression x 2 − 2 x − 80 . This means we want to find two binomials of the form ( x + r ) ( x + s ) such that when we multiply them, we get the original quadratic. The product of r and s must be − 80 , and the sum of r and s must be − 2 .
Finding the Correct Factors We need to find two numbers that multiply to − 80 and add up to − 2 . Let's list some factor pairs of 80:
1 and 80 2 and 40 4 and 20 5 and 16 8 and 10
Since the product is negative, one of the numbers must be negative. Since the sum is − 2 , the larger number must be negative. So we can try:
1 and -80 (sum is -79) 2 and -40 (sum is -38) 4 and -20 (sum is -16) 5 and -16 (sum is -11) 8 and -10 (sum is -2)
We found the pair 8 and -10.
Writing the Factored Form Therefore, the factorization is ( x + 8 ) ( x − 10 ) .
Checking the Answer We can check our answer by expanding the factored form:
( x + 8 ) ( x − 10 ) = x 2 − 10 x + 8 x − 80 = x 2 − 2 x − 80
This matches the original quadratic, so our factorization is correct.
Final Answer The correct factorization of x 2 − 2 x − 80 is ( x + 8 ) ( x − 10 ) . Therefore, the answer is B.
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, suppose you are designing a rectangular garden with an area of x 2 − 2 x − 80 square feet. By factoring this expression into ( x + 8 ) ( x − 10 ) , you determine the dimensions of the garden. If x represents a length, then ( x + 8 ) and ( x − 10 ) represent the width and length of the garden, respectively. Knowing the dimensions helps you plan the layout, fencing, and planting arrangements.
