Which expression in factored form is equivalent to this expression?
$4\left(x^2-2 x\right)-2\left(x^2-3\right)$
A. $\quad 2(x+1)(x+3)$
B. $(2 x-3)(x+1)$
C. $\quad(2 x+3)(x+1)$
D. $2(x-1)(x-3)$
Answer (1)
Expand the given expression: 4 ( x 2 − 2 x ) − 2 ( x 2 − 3 ) = 4 x 2 − 8 x − 2 x 2 + 6 .
Simplify the expression: 2 x 2 − 8 x + 6 .
Factor out the common factor 2: 2 ( x 2 − 4 x + 3 ) .
Factor the quadratic expression: 2 ( x − 1 ) ( x − 3 ) . The correct answer is 2 ( x − 1 ) ( x − 3 ) .
Explanation
Expanding the Expression First, we need to expand the given expression: 4 ( x 2 − 2 x ) − 2 ( x 2 − 3 ) .
Expanding Expanding the expression, we get: 4 x 2 − 8 x − 2 x 2 + 6 .
Simplifying Now, we simplify the expression by combining like terms: ( 4 x 2 − 2 x 2 ) − 8 x + 6 = 2 x 2 − 8 x + 6 .
Factoring out 2 Next, we factor the simplified expression. First, we can factor out a 2: 2 ( x 2 − 4 x + 3 ) .
Factoring Quadratic Now, we need to factor the quadratic expression x 2 − 4 x + 3 . We are looking for two numbers that multiply to 3 and add to -4. These numbers are -1 and -3.
Final Factored Form So, we can factor the quadratic as ( x − 1 ) ( x − 3 ) . Therefore, the fully factored expression is 2 ( x − 1 ) ( x − 3 ) .
Selecting Correct Answer Comparing this with the given options, we see that option D, 2 ( x − 1 ) ( x − 3 ) , is the correct answer.
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures and predict their behavior under different loads. Imagine designing a bridge; the load distribution can be modeled using quadratic equations, and factoring helps determine the critical points where stress is highest, ensuring the bridge's safety and stability. Similarly, in economics, factoring can help analyze cost and revenue functions to find break-even points and optimize profits.
