Factor.
[tex]x^2+12 x+35[/tex]
A. [tex](x+5)(x-7)[/tex]
B. [tex](x+7)(x-5)[/tex]
C. [tex](x+5)(x+7)[/tex]
D. [tex](x-7)(x-5)[/tex]
Answer (2)
Find two numbers that multiply to 35 and add to 12.
The numbers are 5 and 7.
Therefore, the factorization is ( x + 5 ) ( x + 7 ) .
Verify the factorization by expanding ( x + 5 ) ( x + 7 ) to obtain x 2 + 7 x + 5 x + 35 = x 2 + 12 x + 35 .
The final answer is ( x + 5 ) ( x + 7 ) .
Explanation
Understanding the problem We are given a quadratic expression x 2 + 12 x + 35 and asked to factor it. We need to find two numbers that multiply to 35 and add up to 12.
Finding the factors Let's find two numbers that multiply to 35. The pairs of factors of 35 are (1, 35) and (5, 7).
Checking the sum Now, let's check which pair of factors adds up to 12. 1 + 35 = 36, and 5 + 7 = 12. So, the numbers 5 and 7 satisfy both conditions.
Writing the factorization Therefore, the factorization of the quadratic expression is ( x + 5 ) ( x + 7 ) .
Verification To verify, we can expand the factorization: ( x + 5 ) ( x + 7 ) = x 2 + 7 x + 5 x + 35 = x 2 + 12 x + 35 . This matches the original expression.
Final Answer The correct factorization is ( x + 5 ) ( x + 7 ) .
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. In finance, factoring can be used to model investment growth and risk. Understanding factoring helps in optimizing designs, predicting outcomes, and making informed decisions in various fields.
To factor the expression x 2 + 12 x + 35 , we find that the numbers 5 and 7 multiply to 35 and add up to 12, leading to the factorization ( x + 5 ) ( x + 7 ) . Therefore, the correct answer is option C.
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