Factor completely $2 x^2+4 x-2$.
A. $2\left(x^2+2 x-1\right)$
B. $2\left(x^2+2\right)$
C. $2 x\left(x^2+2 x-1\right)$
D. Prime
Answer (2)
Factor out the greatest common factor: 2 x 2 + 4 x − 2 = 2 ( x 2 + 2 x − 1 ) .
Calculate the discriminant of the quadratic x 2 + 2 x − 1 : b 2 − 4 a c = 2 2 − 4 ( 1 ) ( − 1 ) = 8 .
Since the discriminant is not a perfect square, the quadratic cannot be factored further using integers.
The completely factored form is 2 ( x 2 + 2 x − 1 ) .
Explanation
Understanding the Problem We are asked to factor the quadratic expression 2 x 2 + 4 x − 2 completely. This means we want to break it down into its simplest factors.
Factoring out the GCF First, we look for the greatest common factor (GCF) of all the terms in the expression. The GCF of 2 x 2 , 4 x , and − 2 is 2. We factor out the 2 from the expression: 2 x 2 + 4 x − 2 = 2 ( x 2 + 2 x − 1 )
Checking the Discriminant Now, we need to check if the quadratic expression x 2 + 2 x − 1 can be factored further. To do this, we can check its discriminant. The discriminant of a quadratic a x 2 + b x + c is given by b 2 − 4 a c . In our case, a = 1 , b = 2 , and c = − 1 . So the discriminant is: b 2 − 4 a c = ( 2 ) 2 − 4 ( 1 ) ( − 1 ) = 4 + 4 = 8
Final Factorization Since the discriminant (8) is not a perfect square, the quadratic x 2 + 2 x − 1 cannot be factored into integers. Therefore, the completely factored form of the given expression is: 2 ( x 2 + 2 x − 1 )
Examples
Factoring quadratic expressions is a fundamental skill in algebra. It's used in many real-world applications, such as optimizing areas and volumes, modeling projectile motion, and solving engineering problems. For example, if you want to design a rectangular garden with a specific area and perimeter, you might need to factor a quadratic expression to find the dimensions of the garden. Factoring helps simplify complex expressions and makes it easier to solve equations and understand relationships between variables.
To factor 2 x 2 + 4 x − 2 completely, we first factor out the greatest common factor, which is 2, yielding 2 ( x 2 + 2 x − 1 ) . The quadratic x 2 + 2 x − 1 cannot be factored further using integers. Thus, the final answer is option A: 2 ( x 2 + 2 x − 1 ) .
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