Simplify: $2(a+5)-2 a(3 a+4)$
A. $-4 a+2$
B. $-6 a^2-6 a+10$
C. $-8 a^2-6 a+10$
D. $-4 a^2-6 a+5$
Answer (2)
Distribute the terms: 2 ( a + 5 ) = 2 a + 10 and − 2 a ( 3 a + 4 ) = − 6 a 2 − 8 a .
Combine the distributed terms: 2 a + 10 − 6 a 2 − 8 a .
Combine like terms: − 6 a 2 + ( 2 a − 8 a ) + 10 = − 6 a 2 − 6 a + 10 .
The simplified expression is − 6 a 2 − 6 a + 10 .
Explanation
Understanding the Problem We are asked to simplify the expression 2 ( a + 5 ) − 2 a ( 3 a + 4 ) and choose the correct option from the given choices.
Distributing the Terms First, we distribute the terms in the expression. We have
2 ( a + 5 ) = 2 a + 10
and
− 2 a ( 3 a + 4 ) = − 6 a 2 − 8 a .
Combining the Terms Now, we combine the distributed terms:
2 a + 10 − 6 a 2 − 8 a .
Combining Like Terms Next, we combine like terms:
− 6 a 2 + ( 2 a − 8 a ) + 10 = − 6 a 2 − 6 a + 10 .
Comparing with Options Finally, we compare the simplified expression − 6 a 2 − 6 a + 10 with the given options. We see that it matches option B.
Examples
Simplifying algebraic expressions is a fundamental skill in mathematics with applications in various fields. For example, in physics, you might need to simplify an expression representing the force acting on an object. Suppose the force is given by F = 2 ( x + 3 ) − ( 2 x + 1 ) , where x represents the displacement. Simplifying this expression helps in analyzing the behavior of the force with respect to displacement. Similarly, in economics, simplifying expressions can help in analyzing cost functions or revenue functions. For instance, if the cost function is C = 5 ( q + 2 ) − 3 ( q − 1 ) , where q is the quantity produced, simplifying this expression helps in understanding the relationship between cost and quantity.
The expression 2 ( a + 5 ) − 2 a ( 3 a + 4 ) simplifies to − 6 a 2 − 6 a + 10 . Therefore, the correct answer is option B. This involves distributing the terms and combining like terms systematically.
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