One factor of [tex]6 x^2-12 x+18[/tex] is
A. [tex]6(x-6)[/tex]
B. [tex]x^2-2 x+3[/tex]
C. [tex]x+6[/tex]
D. [tex]x^2-12 x+18[/tex]
Answer (2)
The problem requires finding a factor of the quadratic expression 6 x 2 − 12 x + 18 .
Factor out the greatest common divisor (GCD) 6 from the expression: 6 ( x 2 − 2 x + 3 ) .
Compare the factored expression with the given options.
Identify x 2 − 2 x + 3 as a factor.
The correct answer is x 2 − 2 x + 3 .
Explanation
Understanding the Problem The given quadratic expression is 6 x 2 − 12 x + 18 . We need to find one of its factors from the given options.
Factoring out the GCD First, we can factor out the greatest common divisor (GCD) from the quadratic expression. The GCD of the coefficients 6, -12, and 18 is 6. Factoring out 6, we get:
6 x 2 − 12 x + 18 = 6 ( x 2 − 2 x + 3 )
Comparing with the Options Now, we compare the factored expression 6 ( x 2 − 2 x + 3 ) with the given options:
a. 6 ( x − 6 ) b. x 2 − 2 x + 3 c. x + 6 d. x 2 − 12 x + 18
We can see that x 2 − 2 x + 3 is a factor of the given expression.
Final Answer Therefore, the correct answer is b. x 2 − 2 x + 3 .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in various real-world applications. For example, engineers use factoring to analyze the stability of structures, economists use it to model supply and demand curves, and computer scientists use it to optimize algorithms. Understanding how to factor quadratic expressions allows us to simplify complex problems and find solutions more efficiently. In construction, if you need to calculate the area of a rectangular space and you know that the area can be expressed as a quadratic expression, factoring can help you determine the dimensions of the space.
The factor of the quadratic expression 6 x 2 − 12 x + 18 is x 2 − 2 x + 3 , which corresponds to option B. By factoring out the greatest common factor (6), we simplify the expression to identify this factor. Hence, the chosen option is B: x 2 − 2 x + 3 .
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