Factor.
$x^2+12 x+27$
$(x-3)(x-9)$
$(x+3)(x-9)$
$(x+9)(x+3)$
$(x+9)(x-3)$
Answer (1)
To factor the quadratic expression:
Find two numbers that multiply to 27 and add to 12.
Identify the numbers as 3 and 9.
Write the factored form using these numbers: ( x + 3 ) ( x + 9 ) .
The factored form of the quadratic expression is ( x + 3 ) ( x + 9 ) .
Explanation
Understanding the Problem We are given the quadratic expression x 2 + 12 x + 27 and asked to factor it. Factoring involves finding two binomials that, when multiplied together, give us the original quadratic expression.
Finding the Right Numbers To factor the quadratic expression x 2 + 12 x + 27 , we need to find two numbers that multiply to 27 (the constant term) and add up to 12 (the coefficient of the x term).
Identifying the Correct Pair Let's list the factor pairs of 27:
1 and 27 3 and 9
Now, let's check which pair adds up to 12:
1 + 27 = 28 3 + 9 = 12
So, the numbers 3 and 9 satisfy both conditions.
Writing the Factored Form Since the numbers are 3 and 9, the factored form of the quadratic expression is ( x + 3 ) ( x + 9 ) .
Final Answer Therefore, the factored form of x 2 + 12 x + 27 is ( x + 3 ) ( x + 9 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has practical applications in various fields. For example, engineers use factoring to design structures and analyze stress distribution. Imagine you're designing a rectangular garden with an area represented by the quadratic expression x 2 + 12 x + 27 . By factoring this expression into ( x + 3 ) ( x + 9 ) , you determine the dimensions of the garden to be ( x + 3 ) and ( x + 9 ) . This allows you to plan the layout efficiently and optimize the use of space.
