Multiply $(2 x+1)(2 x-3)$
a. $4 x^2-4 x-3$
b. $4 x^2+6 x-3$
c. $4 x^2-4 x+3$
d. $4 x^2+4 x-3$
Answer (1)
Use the distributive property (FOIL) to multiply ( 2 x + 1 ) ( 2 x − 3 ) .
Multiply each term: ( 2 x ) ( 2 x ) + ( 2 x ) ( − 3 ) + ( 1 ) ( 2 x ) + ( 1 ) ( − 3 ) = 4 x 2 − 6 x + 2 x − 3 .
Combine like terms: − 6 x + 2 x = − 4 x .
The final result is 4 x 2 − 4 x − 3 .
Explanation
Understanding the Problem We are given the expression ( 2 x + 1 ) ( 2 x − 3 ) to multiply. Our goal is to expand this expression and identify the correct result from the given options.
Applying the Distributive Property To multiply the two binomials, we use the distributive property, often remembered as the FOIL method (First, Outer, Inner, Last). This means we multiply each term in the first binomial by each term in the second binomial:
( 2 x + 1 ) ( 2 x − 3 ) = ( 2 x ) ( 2 x ) + ( 2 x ) ( − 3 ) + ( 1 ) ( 2 x ) + ( 1 ) ( − 3 )
Simplifying the Expression Now, let's simplify each term:
( 2 x ) ( 2 x ) = 4 x 2 ( 2 x ) ( − 3 ) = − 6 x ( 1 ) ( 2 x ) = 2 x ( 1 ) ( − 3 ) = − 3
So, the expression becomes:
4 x 2 − 6 x + 2 x − 3
Combining Like Terms Next, we combine the like terms (the terms with the same power of x ):
− 6 x + 2 x = − 4 x
So, the simplified expression is:
4 x 2 − 4 x − 3
Identifying the Correct Option Finally, we compare our result, 4 x 2 − 4 x − 3 , with the given options. Option a is 4 x 2 − 4 x − 3 , which matches our result.
Examples
Understanding how to multiply binomials is fundamental in algebra and has many real-world applications. For example, if you're planning a rectangular garden where the length is ( 2 x + 1 ) meters and the width is ( 2 x − 3 ) meters, multiplying these binomials will give you the area of the garden in terms of x . Knowing the area helps you determine how much soil you need, how many plants you can fit, and the overall cost of setting up your garden. This concept extends to various scenarios involving areas, volumes, and other calculations in construction, engineering, and design.
