Factor completely.
[tex]3 x^2+7 x-6[/tex]
A. [tex](x-6)(3 x+1)[/tex]
B. [tex](x-3)(3 x+2)[/tex]
C. [tex](x+3)(3 x-2)[/tex]
D. [tex](x+6)(3 x-1)[/tex]
Answer (2)
Expand each of the given options.
Compare the expanded form of each option with the given quadratic expression.
Identify the option that matches the given quadratic expression.
The correct factorization is ( x + 3 ) ( 3 x − 2 ) .
Explanation
Understanding the Problem We are given the quadratic expression 3 x 2 + 7 x − 6 and asked to factor it completely. We are also given four options to choose from. To solve this, we will expand each option and see which one matches the given quadratic expression.
Expanding the Options Let's expand each of the options:
Option A: ( x − 6 ) ( 3 x + 1 ) = 3 x 2 + x − 18 x − 6 = 3 x 2 − 17 x − 6 . This does not match the given expression.
Option B: ( x − 3 ) ( 3 x + 2 ) = 3 x 2 + 2 x − 9 x − 6 = 3 x 2 − 7 x − 6 . This does not match the given expression.
Option C: ( x + 3 ) ( 3 x − 2 ) = 3 x 2 − 2 x + 9 x − 6 = 3 x 2 + 7 x − 6 . This matches the given expression.
Option D: ( x + 6 ) ( 3 x − 1 ) = 3 x 2 − x + 18 x − 6 = 3 x 2 + 17 x − 6 . This does not match the given expression.
Finding the Correct Factorization Comparing the expanded forms with the given quadratic expression 3 x 2 + 7 x − 6 , we find that option C, ( x + 3 ) ( 3 x − 2 ) , matches the given expression.
Final Answer Therefore, the correct factorization of 3 x 2 + 7 x − 6 is ( x + 3 ) ( 3 x − 2 ) .
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. Architects use factoring to create aesthetically pleasing and structurally sound buildings. Financial analysts use factoring to model and predict market trends. In general, factoring helps simplify complex problems and find solutions more efficiently.
The quadratic expression 3 x 2 + 7 x − 6 can be factored as ( x + 3 ) ( 3 x − 2 ) . This is confirmed by expanding the options provided and comparing them to the original expression. Therefore, the correct answer is option C.
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