Factor: $(x^2-11 x+24)$
A. $(x-8)(x-3)$
B. $(x+8)(x+3)$
C. $(x+6)(x+4)$
D. $(x-6)(x-4)$
Answer (1)
To factor the quadratic expression x 2 − 11 x + 24 , we need to find two numbers that multiply to 24 and add up to -11. These numbers are -3 and -8. Therefore, the factored form is ( x − 8 ) ( x − 3 ) . The correct option is A.
Explanation
Understanding the Problem We are given the quadratic expression x 2 − 11 x + 24 and asked to factor it. Factoring a quadratic means expressing it as a product of two binomials. We need to find two numbers that multiply to 24 (the constant term) and add up to -11 (the coefficient of the x term).
Finding the Correct Factors Let's consider the factors of 24: 1 and 24, 2 and 12, 3 and 8, 4 and 6. Since the middle term is -11 and the last term is +24, we need two negative numbers that multiply to 24 and add to -11. The numbers -3 and -8 satisfy these conditions since ( − 3 ) × ( − 8 ) = 24 and ( − 3 ) + ( − 8 ) = − 11 .
Writing the Factored Form Therefore, the factored form of the quadratic expression is ( x − 3 ) ( x − 8 ) . We can also write it as ( x − 8 ) ( x − 3 ) , since multiplication is commutative.
Selecting the Correct Option Comparing our factored form with the given options, we see that option A, ( x − 8 ) ( x − 3 ) , matches our result.
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 with an area represented by x 2 − 11 x + 24 . By factoring this expression into ( x − 8 ) ( x − 3 ) , you determine the possible dimensions of the garden, which helps in planning the layout and fencing requirements. This skill is also crucial in physics for solving projectile motion problems and in economics for modeling cost and revenue functions.
