The expression $(4 c-3 d)(3 c+d)$ is equivalent to:
A. $12 c^2-13 c d-3 d^2$
B. $12 c^2-13 c d+3 d^2$
C. $12 c^2-5 c d-3 d^2$
D. $12 c^2-5 c d+3 d^2$
E. $12 c ^2-3 d^2$
Answer (1)
To find the equivalent expression, we expand ( 4 c − 3 d ) ( 3 c + d ) using the distributive property:
Multiply 4 c by 3 c and d to get 12 c 2 + 4 c d .
Multiply − 3 d by 3 c and d to get − 9 c d − 3 d 2 .
Combine all terms: 12 c 2 + 4 c d − 9 c d − 3 d 2 .
Simplify by combining like terms: 12 c 2 − 5 c d − 3 d 2 .
Thus, the equivalent expression is 12 c 2 − 5 c d − 3 d 2 .
Explanation
Understanding the Problem We are given the expression ( 4 c − 3 d ) ( 3 c + d ) and asked to find an equivalent expression from the given options.
Expanding the Expression To find the equivalent expression, we need to expand the given expression using the distributive property (also known as the FOIL method). This involves multiplying each term in the first parentheses by each term in the second parentheses.
Applying the Distributive Property Let's expand the expression step by step:
First, multiply 4 c by both terms in the second parentheses: 4 c × 3 c = 12 c 2 4 c × d = 4 c d
Next, multiply − 3 d by both terms in the second parentheses: − 3 d × 3 c = − 9 c d − 3 d × d = − 3 d 2
Now, combine all the terms: 12 c 2 + 4 c d − 9 c d − 3 d 2
Simplifying the Expression Now, we simplify the expression by combining like terms. The like terms are 4 c d and − 9 c d :
4 c d − 9 c d = − 5 c d
So, the simplified expression is: 12 c 2 − 5 c d − 3 d 2
Comparing with the Options Finally, we compare the simplified expression 12 c 2 − 5 c d − 3 d 2 with the given options. We see that it matches option C.
Final Answer Therefore, the expression ( 4 c − 3 d ) ( 3 c + d ) is equivalent to 12 c 2 − 5 c d − 3 d 2 .
Examples
Understanding how to expand and simplify algebraic expressions like this is fundamental in many areas, such as physics and engineering. For example, when calculating the area of a rectangular garden where the sides are expressed as algebraic expressions, you would need to expand and simplify to find the total area. Similarly, in physics, when dealing with forces or motion, you might encounter expressions that need simplification to solve for unknown variables. This skill provides a foundation for solving more complex problems in various fields.
