Which of the following are the factors of $m^2-14 m+48$?
A) $(m-12)(m+4)$
B) $(m-12)(m-4)$
C) $(m-6)(m-8)$
D) $(m+6)(m+8)$
Answer (1)
We need to factor the quadratic expression m 2 − 14 m + 48 .
Find two numbers that multiply to 48 and add up to -14. These numbers are -6 and -8.
Write the factored form as ( m − 6 ) ( m − 8 ) .
The correct answer is ( m − 6 ) ( m − 8 ) .
Explanation
Understanding the Problem We are given the quadratic expression m 2 − 14 m + 48 and asked to find its factors. Factoring a quadratic expression involves finding two binomials that, when multiplied together, give us the original quadratic expression.
Finding the Factors To factor the quadratic expression m 2 − 14 m + 48 , we need to find two numbers that multiply to 48 (the constant term) and add up to -14 (the coefficient of the m term). Let's list the factor pairs of 48:
1 and 48 2 and 24 3 and 16 4 and 12 6 and 8
Since we need the two numbers to add up to -14, we should consider the negative pairs:
-1 and -48 -2 and -24 -3 and -16 -4 and -12 -6 and -8
Identifying the Correct Pair From the list of negative factor pairs, we can see that -6 and -8 add up to -14: − 6 + ( − 8 ) = − 14 . Also, − 6 × − 8 = 48 . Therefore, the two numbers we are looking for are -6 and -8.
Writing the Factored Form Now we can write the factored form of the quadratic expression using these two numbers: ( m − 6 ) ( m − 8 ) .
Selecting the Correct Option Comparing our factored form ( m − 6 ) ( m − 8 ) with the given options, we see that option C matches our result.
Final Answer Therefore, the factors of m 2 − 14 m + 48 are ( m − 6 ) ( m − 8 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, suppose you are designing a rectangular garden and you know the area of the garden is given by the expression m 2 − 14 m + 48 , where m is the length of one side. By factoring this expression into ( m − 6 ) ( m − 8 ) , you can determine the possible dimensions of the garden. If you want the garden to have an area of 0, you can set each factor to zero and solve for m , which gives you the possible lengths of the sides. This helps in planning and optimizing the use of space in various design and construction projects.
