Which statement about $4 x^2+19 x-5$ is true?
A. One of the factors is $(x-4)$.
B. One of the factors is $(4 x+1)$.
C. One of the factors is $(4 x-5)$.
D. One of the factors is $(x+5)$.
Answer (1)
Factor the quadratic expression 4 x 2 + 19 x − 5 by finding two numbers that multiply to − 20 and add to 19 , which are 20 and − 1 .
Rewrite the middle term and factor by grouping: 4 x 2 + 20 x − x − 5 = 4 x ( x + 5 ) − 1 ( x + 5 ) = ( 4 x − 1 ) ( x + 5 ) .
Identify the factors as ( 4 x − 1 ) and ( x + 5 ) .
Determine that the correct statement is that ( x + 5 ) is one of the factors. ( x + 5 )
Explanation
Problem Analysis We are given the quadratic expression 4 x 2 + 19 x − 5 and asked to identify the correct factor from the options: ( x − 4 ) , ( 4 x + 1 ) , ( 4 x − 5 ) , and ( x + 5 ) .
Factoring the Quadratic We can factor the quadratic expression 4 x 2 + 19 x − 5 . We look for two numbers that multiply to 4 × − 5 = − 20 and add up to 19 . These numbers are 20 and − 1 . So we can rewrite the middle term as 19 x = 20 x − x . Therefore, we have:
4 x 2 + 19 x − 5 = 4 x 2 + 20 x − x − 5
Factoring by Grouping Now we factor by grouping:
4 x 2 + 20 x − x − 5 = 4 x ( x + 5 ) − 1 ( x + 5 ) = ( 4 x − 1 ) ( x + 5 )
Identifying the Correct Factor Thus, the factors of 4 x 2 + 19 x − 5 are ( 4 x − 1 ) and ( x + 5 ) . Comparing these factors with the given options, we see that ( x + 5 ) is one of the factors.
Verification of Factors Alternatively, we can test each of the given factors to see if they divide the quadratic expression. If ( x − 4 ) is a factor, then x = 4 is a root of the quadratic. Substituting x = 4 into the quadratic gives 4 ( 4 ) 2 + 19 ( 4 ) − 5 = 4 ( 16 ) + 76 − 5 = 64 + 76 − 5 = 135 = 0 . So ( x − 4 ) is not a factor.
If ( 4 x + 1 ) is a factor, then x = − 4 1 is a root of the quadratic. Substituting x = − 4 1 into the quadratic gives 4 ( − 4 1 ) 2 + 19 ( − 4 1 ) − 5 = 4 ( 16 1 ) − 4 19 − 5 = 4 1 − 4 19 − 4 20 = 4 1 − 19 − 20 = 4 − 38 = − 2 19 = 0 . So ( 4 x + 1 ) is not a factor.
If ( 4 x − 5 ) is a factor, then x = 4 5 is a root of the quadratic. Substituting x = 4 5 into the quadratic gives 4 ( 4 5 ) 2 + 19 ( 4 5 ) − 5 = 4 ( 16 25 ) + 4 95 − 5 = 4 25 + 4 95 − 4 20 = 4 25 + 95 − 20 = 4 100 = 25 = 0 . So ( 4 x − 5 ) is not a factor.
If ( x + 5 ) is a factor, then x = − 5 is a root of the quadratic. Substituting x = − 5 into the quadratic gives 4 ( − 5 ) 2 + 19 ( − 5 ) − 5 = 4 ( 25 ) − 95 − 5 = 100 − 95 − 5 = 0 . So ( x + 5 ) is a factor.
Final Answer Therefore, the correct statement is that ( x + 5 ) is one of the factors of 4 x 2 + 19 x − 5 .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in various real-world applications. For instance, engineers use factoring to design structures and calculate stress and strain. Consider a bridge design where the load distribution can be modeled by a quadratic equation. By factoring this equation, engineers can determine the critical points where the stress is highest, ensuring the bridge's stability and safety. Similarly, in physics, factoring helps solve problems related to projectile motion, where the trajectory of an object can be described by a quadratic equation. Factoring allows physicists to find the time and distance at which the projectile reaches its maximum height or lands, which is crucial for applications like aiming artillery or designing sports equipment.
