The factorization of [tex]x^2-6 x-72[/tex] is:
[tex](x+6)(x+12)[/tex]
[tex](x-6)(x-12)[/tex]
[tex](x-6)(x+12)[/tex]
[tex](x-12)(x+6)[/tex]
Answer (2)
Find two numbers whose product is -72 and sum is -6.
The numbers are 6 and -12.
The factorization is ( x + 6 ) ( x − 12 ) .
The factorization of x 2 − 6 x − 72 is ( x − 12 ) ( x + 6 ) .
Explanation
Understanding the Problem We are given the quadratic expression x 2 − 6 x − 72 and asked to find its factorization from the given options.
Finding the Factors We need to find two numbers that multiply to -72 and add up to -6. Let's list the factor pairs of -72:
(1, -72), (2, -36), (3, -24), (4, -18), (6, -12), (8, -9), (-1, 72), (-2, 36), (-3, 24), (-4, 18), (-6, 12), (-8, 9).
Checking the Sums Now, let's check which of these pairs adds up to -6:
1 + (-72) = -71 2 + (-36) = -34 3 + (-24) = -21 4 + (-18) = -14 6 + (-12) = -6 8 + (-9) = -1 -1 + 72 = 71 -2 + 36 = 34 -3 + 24 = 21 -4 + 18 = 14 -6 + 12 = 6 -8 + 9 = 1
Determining the Factorization The pair (6, -12) adds up to -6. Therefore, the factorization of the quadratic expression is ( x + 6 ) ( x − 12 ) or equivalently ( x − 12 ) ( x + 6 ) .
Final Answer The correct factorization is ( x − 12 ) ( x + 6 ) .
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, ensuring stability and optimal use of materials. Imagine designing a rectangular garden where you know the area and need to determine the possible dimensions. Factoring the quadratic equation representing the area helps you find the possible lengths and widths, allowing you to plan your garden effectively. This skill also helps in optimizing various processes, from manufacturing to logistics, by finding the most efficient solutions to problems involving quadratic relationships.
The factorization of the expression x 2 − 6 x − 72 is ( x − 12 ) ( x + 6 ) . This is determined by finding two numbers that multiply to -72 and add to -6, which are 6 and -12. Therefore, the answer is ( x − 12 ) ( x + 6 ) .
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