Factor $2 x^2+7 x-49$
A. $(x-7)(2 x-7)$
B. $(2 x+7)(x+7)$
C. $(x+7)(2 x-7)$
D. $(2 x+7)(x-7)$
Answer (1)
We are asked to factor the quadratic expression 2 x 2 + 7 x − 49 .
We can expand each of the given options and see which one matches the original quadratic expression.
After expanding each option, we find that ( x + 7 ) ( 2 x − 7 ) = 2 x 2 + 7 x − 49 .
Therefore, the correct factorization is ( x + 7 ) ( 2 x − 7 ) .
Explanation
Understanding the Problem We are given a quadratic expression 2 x 2 + 7 x − 49 and asked to factor it. The possible factors are given as options.
Solution Plan We can factor the quadratic expression by finding two binomials of the form ( a x + b ) ( c x + d ) such that a c = 2 , a d + b c = 7 , and b d = − 49 . Alternatively, we can expand each of the given options and see which one matches the original quadratic expression.
Expanding the Options Let's expand the given options:
Option 1: ( x − 7 ) ( 2 x − 7 ) = 2 x 2 − 7 x − 14 x + 49 = 2 x 2 − 21 x + 49 . This does not match.
Option 2: ( 2 x + 7 ) ( x + 7 ) = 2 x 2 + 14 x + 7 x + 49 = 2 x 2 + 21 x + 49 . This does not match.
Option 3: ( x + 7 ) ( 2 x − 7 ) = 2 x 2 − 7 x + 14 x − 49 = 2 x 2 + 7 x − 49 . This matches.
Option 4: ( 2 x + 7 ) ( x − 7 ) = 2 x 2 − 14 x + 7 x − 49 = 2 x 2 − 7 x − 49 . This does not match.
Conclusion Therefore, the correct factorization is ( x + 7 ) ( 2 x − 7 ) .
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 you are designing a bridge, and you need to calculate the forces acting on it. By factoring quadratic expressions, you can determine the optimal placement of supports to ensure the bridge's stability. Similarly, in physics, factoring is used to solve equations related to projectile motion and other phenomena. This algebraic technique allows for efficient problem-solving and accurate predictions in various scientific and engineering fields.
