Factor:
[tex]x^2+12 x+20[/tex]
Answer (1)
To factor the quadratic expression:
Find two numbers that multiply to 20 and add up to 12.
Identify the numbers as 2 and 10.
Write the factored form as ( x + 2 ) ( x + 10 ) .
The factored form is ( x + 2 ) ( x + 10 ) .
Explanation
Understanding the Problem We are asked to factor the quadratic expression x 2 + 12 x + 20 . Our goal is to find two binomials that, when multiplied together, give us the original quadratic expression.
Finding the Numbers To factor the quadratic expression x 2 + 12 x + 20 , we need to find two numbers that multiply to 20 (the constant term) and add up to 12 (the coefficient of the x term). Let's call these two numbers m and n . We are looking for m and n such that:
m n = 20 m + n = 12
Identifying the Correct Pair Let's list the pairs of factors of 20:
1 and 20 2 and 10 4 and 5
Now, let's check which pair adds up to 12:
1 + 20 = 21 2 + 10 = 12 4 + 5 = 9
The pair 2 and 10 satisfies both conditions. So, m = 2 and n = 10 .
Writing the Factored Form Now that we have found the two numbers, we can write the factored form of the quadratic expression as ( x + m ) ( x + n ) . Substituting m = 2 and n = 10 , we get:
( x + 2 ) ( x + 10 )
Verifying the Factorization To verify our factorization, we can expand ( x + 2 ) ( x + 10 ) :
( x + 2 ) ( x + 10 ) = x ( x + 10 ) + 2 ( x + 10 ) = x 2 + 10 x + 2 x + 20 = x 2 + 12 x + 20
Since the expanded form matches the original quadratic expression, our factorization is correct.
Final Answer Therefore, the factored form of x 2 + 12 x + 20 is ( x + 2 ) ( x + 10 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, suppose you want to build a rectangular garden with an area of x 2 + 12 x + 20 square feet. By factoring this expression into ( x + 2 ) ( x + 10 ) , you determine that the dimensions of the garden could be ( x + 2 ) feet and ( x + 10 ) feet. This allows you to plan the layout of your garden based on the available space and desired area.
