Select the correct answer.
Which expression is equivalent to the given expression?
$3 x^2+5 x-7\left(x^2+4\right)$
A. $-4 x^2+5 x-4$
B. $x^2+28$
C. $x^2+4$
D. $-4 x^2+5 x-28$
Answer (1)
Distribute the -7: − 7 ( x 2 + 4 ) = − 7 x 2 − 28
Substitute back into the original expression: 3 x 2 + 5 x − 7 x 2 − 28
Combine like terms: ( 3 x 2 − 7 x 2 ) + 5 x − 28 = − 4 x 2 + 5 x − 28
The equivalent expression is: − 4 x 2 + 5 x − 28
Explanation
Understanding the Problem We are given the expression 3 x 2 + 5 x − 7 ( x 2 + 4 ) and asked to find an equivalent expression from the given options.
Distributing the -7 First, we need to distribute the − 7 to the terms inside the parenthesis: − 7 ( x 2 + 4 ) = − 7 × x 2 − 7 × 4 = − 7 x 2 − 28
Substituting Back Now, we substitute this back into the original expression: 3 x 2 + 5 x − 7 ( x 2 + 4 ) = 3 x 2 + 5 x − 7 x 2 − 28
Combining Like Terms (x^2 terms) Next, we combine like terms. We combine the x 2 terms: 3 x 2 − 7 x 2 = ( 3 − 7 ) x 2 = − 4 x 2
Simplified Expression So the expression becomes: − 4 x 2 + 5 x − 28
Finding the Correct Option Comparing this to the given options, we see that option D, − 4 x 2 + 5 x − 28 , is the correct answer.
Examples
Simplifying algebraic expressions is a fundamental skill in mathematics and has numerous real-world applications. For instance, consider a scenario where a company's profit is modeled by a quadratic expression. By simplifying the expression, the company can easily analyze and predict its profit based on different variables, such as production costs and sales volume. This allows for better decision-making and strategic planning.
