Select the correct answer.
What is the complete factorization of $x^2+4 x-45 ?$
A. $(x+15)(x-3)$
B. $(x-9)(x+5)$
C. $(x+9)(x-5)$
D. $(x-15)(x+3)$
Answer (1)
Find two numbers that multiply to -45 and add to 4.
The numbers are -5 and 9.
Write the factorization as ( x − 5 ) ( x + 9 ) .
The complete factorization is ( x + 9 ) ( x − 5 ) .
Explanation
Understanding the Problem We are given the quadratic expression x 2 + 4 x − 45 and asked to find its complete factorization. This means we want to rewrite the expression as a product of two binomials of the form ( x + a ) ( x + b ) , where a and b are constants.
Finding the Factors To factor the quadratic expression x 2 + 4 x − 45 , we need to find two numbers that multiply to -45 (the constant term) and add up to 4 (the coefficient of the x term).
Listing Factor Pairs Let's list the pairs of factors of -45:
1 and -45 -1 and 45 3 and -15 -3 and 15 5 and -9 -5 and 9
Checking the Sum Now, let's check which of these pairs adds up to 4:
1 + (-45) = -44 -1 + 45 = 44 3 + (-15) = -12 -3 + 15 = 12 5 + (-9) = -4 -5 + 9 = 4
The pair -5 and 9 adds up to 4.
Writing the Factorization Therefore, the factorization of the quadratic expression is ( x − 5 ) ( x + 9 ) .
Selecting the Correct Answer Comparing our result with the given options, we see that option C, ( x + 9 ) ( x − 5 ) , matches our factorization.
Final Answer Thus, the complete factorization of x 2 + 4 x − 45 is ( x + 9 ) ( x − 5 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, engineers use factoring to analyze the stability of structures, economists use it to model supply and demand curves, and computer scientists use it to optimize algorithms. Imagine you are designing a rectangular garden with an area represented by the expression x 2 + 4 x − 45 . By factoring this expression into ( x + 9 ) ( x − 5 ) , you determine the dimensions of the garden, ensuring it fits perfectly in your backyard. This skill is also crucial in physics for solving projectile motion problems, where you need to find the time when an object hits the ground, which often involves solving quadratic equations.
