Which of the following represents the factorization of the trinomial below?
[tex]x^2-12 x+20[/tex]
A. [tex](x-4)(x-5)[/tex]
B. [tex](x-2)(x-10)[/tex]
C. [tex](x+4)(x-5)[/tex]
D. [tex](x-4)(x-5)[/tex]
Answer (1)
We need to factor the quadratic trinomial x 2 − 12 x + 20 .
Find two numbers that multiply to 20 and add up to -12. The numbers are -2 and -10.
Write the factorization using these numbers: ( x − 2 ) ( x − 10 ) .
The correct factorization is ( x − 2 ) ( x − 10 ) .
Explanation
Understanding the Problem We are given the quadratic trinomial x 2 − 12 x + 20 and asked to find its factorization from the given options.
Finding the Factors To factor the quadratic trinomial x 2 − 12 x + 20 , we need to find two numbers that multiply to 20 and add up to -12.
Listing Factor Pairs Let's list the factor pairs of 20: (1, 20), (2, 10), (4, 5). Since the middle term is negative and the last term is positive, we consider the negative factor pairs: (-1, -20), (-2, -10), (-4, -5).
Checking the Sum Now, let's check which pair adds up to -12:
-1 + (-20) = -21
-2 + (-10) = -12
-4 + (-5) = -9 The pair -2 and -10 adds up to -12.
Final Factorization Therefore, the factorization of the quadratic trinomial x 2 − 12 x + 20 is ( x − 2 ) ( x − 10 ) .
Selecting the Correct Option The correct answer is B. ( x − 2 ) ( x − 10 ) .
Examples
Factoring quadratic trinomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures and solve problems related to stress and strain. Imagine you are designing a rectangular garden with an area represented by the trinomial x 2 − 12 x + 20 . By factoring this trinomial into ( x − 2 ) ( x − 10 ) , you determine the dimensions of the garden to be ( x − 2 ) and ( x − 10 ) . This allows you to plan the layout and efficiently use the available space.
