Which statement about $-2 h^2-15 h-7$ is true?
A. One of the factors is $(h+2)$.
B. One of the factors is $(3 h-2)$.
C. One of the factors is $(2 h+1)$.
D. One of the factors is $(h-7)$.
Answer (1)
Factor the quadratic expression − 2 h 2 − 15 h − 7 as − ( 2 h + 1 ) ( h + 7 ) .
Compare the factors with the given options.
Identify the matching factor as ( 2 h + 1 ) .
Conclude that the statement 'One of the factors is ( 2 h + 1 ) ' is true.
Explanation
Analyze the problem We are given the quadratic expression − 2 h 2 − 15 h − 7 and asked to identify the correct factor from the options: ( h + 2 ) , ( 3 h − 2 ) , ( 2 h + 1 ) , and ( h − 7 ) . To do this, we will factor the quadratic expression and then compare the factors with the given options.
Factor the quadratic expression First, let's factor the quadratic expression − 2 h 2 − 15 h − 7 . We can rewrite this expression as − ( 2 h 2 + 15 h + 7 ) . Now, we need to find two numbers that multiply to 2 × 7 = 14 and add up to 15 . These numbers are 1 and 14 . So, we can rewrite the expression as:
− ( 2 h 2 + 14 h + h + 7 )
Now, we can factor by grouping:
− ( 2 h ( h + 7 ) + 1 ( h + 7 ))
− ( 2 h + 1 ) ( h + 7 )
So, the factors of the quadratic expression are − ( 2 h + 1 ) and ( h + 7 ) , or ( 2 h + 1 ) and − ( h + 7 ) .
Compare the factors Now, let's compare the factors we found with the given options:
( h + 2 ) is not a factor.
( 3 h − 2 ) is not a factor.
( 2 h + 1 ) is a factor.
( h − 7 ) is not a factor.
Identify the correct factor Therefore, the correct statement is that one of the factors is ( 2 h + 1 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, engineers use factoring to design structures and calculate stress and strain. Architects use factoring to create blueprints and ensure that buildings are stable. Financial analysts use factoring to model investments and predict market trends. In general, factoring helps to simplify complex problems and find solutions more efficiently. For instance, if you are trying to determine the dimensions of a rectangular garden with a specific area, you might use factoring to find the possible lengths and widths.
