Factor.
[tex]x^2+14 x+48[/tex]
A. [tex](x+6)(x-8)[/tex]
B. [tex](x+8)(x-6)[/tex]
C. [tex](x-8)(x-6)[/tex]
D. [tex](x+6)(x+8)[/tex]
Answer (2)
To factor the quadratic expression x 2 + 14 x + 48 :
Find two numbers that multiply to 48 and add to 14.
Identify the factor pair (6, 8) that satisfies these conditions.
Write the factored form using these numbers: ( x + 6 ) ( x + 8 ) .
The factored form of the quadratic expression is ( x + 6 ) ( x + 8 ) .
Explanation
Understanding the Problem We are given the quadratic expression x 2 + 14 x + 48 and asked to factor it. Factoring a quadratic expression means finding two binomials that, when multiplied together, give us the original quadratic expression.
Finding the Factors We need to find two numbers that multiply to 48 (the constant term) and add up to 14 (the coefficient of the x term). Let's list the pairs of factors of 48:
(1, 48) (2, 24) (3, 16) (4, 12) (6, 8)
Identifying the Correct Pair Now, let's check which of these pairs adds up to 14:
1 + 48 = 49 2 + 24 = 26 3 + 16 = 19 4 + 12 = 16 6 + 8 = 14
The pair (6, 8) adds up to 14.
Writing the Factored Form Since the numbers 6 and 8 multiply to 48 and add to 14, the factored form of the quadratic expression is ( x + 6 ) ( x + 8 ) .
Final Answer Therefore, the correct factorization of x 2 + 14 x + 48 is ( x + 6 ) ( x + 8 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, engineers use factoring to design structures and solve problems related to stress and strain. Imagine you are designing a rectangular garden with an area represented by the quadratic expression x 2 + 14 x + 48 . By factoring this expression into ( x + 6 ) ( x + 8 ) , you determine that the dimensions of the garden are ( x + 6 ) and ( x + 8 ) . This allows you to plan the layout of the garden efficiently and ensure that it meets your desired area.
The factored form of the quadratic expression x 2 + 14 x + 48 is ( x + 6 ) ( x + 8 ) . This is derived by finding two numbers that multiply to 48 and add to 14. The correct option among the given choices is D: ( x + 6 ) ( x + 8 ) .
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