Factor: [tex]x^2+2 x-48[/tex]

A. [tex](x+2)(x-8)[/tex]
B. [tex](x-8)(x+6)[/tex]
C. [tex](x+8)(x-2)[/tex]
D. [tex](x+8)(x-6)[/tex]

Find two numbers that multiply to -48 and add to 2.
The numbers are -6 and 8.
Write the factored form as ( x − 6 ) ( x + 8 ) or ( x + 8 ) ( x − 6 ) .
The factored form is ( x + 8 ) ( x − 6 ) ​ .

Explanation

Understanding the Problem We are given the quadratic expression x 2 + 2 x − 48 and asked to factor it. This means we want to find two binomials of the form ( x + a ) ( x + b ) such that when we multiply them, we get the original quadratic expression.

Finding the Correct Factors To factor the quadratic expression x 2 + 2 x − 48 , we need to find two numbers that multiply to -48 and add to 2. Let's list the factor pairs of -48:

1 and -48

-1 and 48

2 and -24

-2 and 24

3 and -16

-3 and 16

4 and -12

-4 and 12

6 and -8

-6 and 8

Identifying the Correct Pair Now, we check which of these pairs add up to 2:

1 + ( − 48 ) = − 47

− 1 + 48 = 47

2 + ( − 24 ) = − 22

− 2 + 24 = 22

3 + ( − 16 ) = − 13

− 3 + 16 = 13

4 + ( − 12 ) = − 8

− 4 + 12 = 8

6 + ( − 8 ) = − 2

− 6 + 8 = 2


The pair -6 and 8 add up to 2.

Writing the Factored Form Therefore, the factored form of the quadratic expression is ( x − 6 ) ( x + 8 ) or ( x + 8 ) ( x − 6 ) .

Final Answer Comparing our factored form with the given options, we see that the correct option is ( x + 8 ) ( x − 6 ) .

Conclusion Thus, the factored form of x 2 + 2 x − 48 is ( x + 8 ) ( x − 6 ) .


Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, if you are designing a rectangular garden and you know the area is represented by the expression x 2 + 2 x − 48 , where x is a variable related to the dimensions, factoring the expression helps you determine the possible lengths and widths of the garden. In this case, the dimensions could be ( x + 8 ) and ( x − 6 ) . This skill is also useful in physics for solving projectile motion problems, where the height of an object can be modeled by a quadratic equation.

Answered by GinnyAnswer | 2025-07-05