Factor.
$x^2+8 x-33$
A. $(x+11)(x+3)$
B. $(x-3)(x+11)$
C. $(x-3)(x-11)$
D. $(x-11)(x+3)$
Answer (1)
Expand each of the given options.
Compare the expanded form with the original quadratic expression x 2 + 8 x − 33 .
Select the option that matches the original quadratic expression.
The correct factorization is ( x − 3 ) ( x + 11 ) .
Explanation
Understanding the Problem We are asked to factor the quadratic expression x 2 + 8 x − 33 . This means we need to find two binomials of the form ( x + a ) ( x + b ) such that when multiplied, they give us the original quadratic expression. We are given four possible factorizations, and we need to determine which one is correct.
Solution Strategy To find the correct factorization, we will expand each of the given options and compare the result with the original quadratic expression x 2 + 8 x − 33 .
Expanding Option 1 Let's expand the first option, ( x + 11 ) ( x + 3 ) : ( x + 11 ) ( x + 3 ) = x ( x + 3 ) + 11 ( x + 3 ) = x 2 + 3 x + 11 x + 33 = x 2 + 14 x + 33 This does not match the original expression x 2 + 8 x − 33 .
Expanding Option 2 Now, let's expand the second option, ( x − 3 ) ( x + 11 ) : ( x − 3 ) ( x + 11 ) = x ( x + 11 ) − 3 ( x + 11 ) = x 2 + 11 x − 3 x − 33 = x 2 + 8 x − 33 This matches the original expression x 2 + 8 x − 33 .
Expanding Option 3 Let's expand the third option, ( x − 3 ) ( x − 11 ) : ( x − 3 ) ( x − 11 ) = x ( x − 11 ) − 3 ( x − 11 ) = x 2 − 11 x − 3 x + 33 = x 2 − 14 x + 33 This does not match the original expression x 2 + 8 x − 33 .
Expanding Option 4 Finally, let's expand the fourth option, ( x − 11 ) ( x + 3 ) : ( x − 11 ) ( x + 3 ) = x ( x + 3 ) − 11 ( x + 3 ) = x 2 + 3 x − 11 x − 33 = x 2 − 8 x − 33 This does not match the original expression x 2 + 8 x − 33 .
Conclusion After expanding all the options, we found that the second option, ( x − 3 ) ( x + 11 ) , matches the original quadratic expression x 2 + 8 x − 33 . Therefore, the correct factorization is ( x − 3 ) ( x + 11 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures and predict their behavior under different loads. Imagine designing a bridge; the load and stress on the bridge can often be modeled using quadratic equations. By factoring these equations, engineers can determine the critical points where the stress is highest, ensuring the bridge is built to withstand those forces and remain safe. This ensures structural integrity and prevents potential failures.
