Which expression is equivalent to [tex]$\left(3 x^2-7\right)$[/tex]?
[tex]$\left(2 x^2-11\right)-\left(x^2+4\right)$[/tex]
[tex]$\left(5 x^2-6\right)-\left(2 x^2-1\right)$[/tex]
[tex]$\left(10 x^2-4\right)-\left(7 x^2+3\right)$[/tex]
[tex]$\left(15 x^2+8\right)-\left(18 x^2+1\right)$[/tex]
Answer (1)
Simplify each of the given expressions.
Compare the simplified expressions to 3 x 2 − 7 .
Identify that ( 10 x 2 − 4 ) − ( 7 x 2 + 3 ) simplifies to 3 x 2 − 7 .
The equivalent expression is ( 10 x 2 − 4 ) − ( 7 x 2 + 3 ) .
Explanation
Problem Analysis We are given the expression 3 x 2 − 7 and asked to find an equivalent expression from the list:
( 2 x 2 − 11 ) − ( x 2 + 4 )
( 5 x 2 − 6 ) − ( 2 x 2 − 1 )
( 10 x 2 − 4 ) − ( 7 x 2 + 3 )
( 15 x 2 + 8 ) − ( 18 x 2 + 1 )
Simplifying the Expressions Let's simplify each of the given expressions:
( 2 x 2 − 11 ) − ( x 2 + 4 ) = 2 x 2 − 11 − x 2 − 4 = ( 2 x 2 − x 2 ) + ( − 11 − 4 ) = x 2 − 15
( 5 x 2 − 6 ) − ( 2 x 2 − 1 ) = 5 x 2 − 6 − 2 x 2 + 1 = ( 5 x 2 − 2 x 2 ) + ( − 6 + 1 ) = 3 x 2 − 5
( 10 x 2 − 4 ) − ( 7 x 2 + 3 ) = 10 x 2 − 4 − 7 x 2 − 3 = ( 10 x 2 − 7 x 2 ) + ( − 4 − 3 ) = 3 x 2 − 7
( 15 x 2 + 8 ) − ( 18 x 2 + 1 ) = 15 x 2 + 8 − 18 x 2 − 1 = ( 15 x 2 − 18 x 2 ) + ( 8 − 1 ) = − 3 x 2 + 7 = 7 − 3 x 2
Finding the Equivalent Expression Comparing the simplified expressions to the given expression 3 x 2 − 7 , we see that expression 3 simplifies to 3 x 2 − 7 .
Therefore, the expression equivalent to 3 x 2 − 7 is ( 10 x 2 − 4 ) − ( 7 x 2 + 3 ) .
Final Answer The equivalent expression is ( 10 x 2 − 4 ) − ( 7 x 2 + 3 ) .
Examples
Understanding how to simplify and manipulate algebraic expressions is crucial in various fields. For instance, in physics, you might use these skills to rearrange equations to solve for unknown variables. In engineering, simplifying expressions can help optimize designs or analyze system performance. Even in economics, you might use algebraic manipulation to model and predict market trends. The ability to simplify expressions allows us to solve complex problems efficiently and accurately.
