Factor.
$x^2-11 x+18$
$(x-2)(x+9)$
$(x-9)(x-2)$
$(x+2)(x+9)$
$(x-9)(x+2)$
Answer (1)
We need to factor the quadratic expression x 2 − 11 x + 18 .
Find two numbers that multiply to 18 and add up to -11. These numbers are -2 and -9.
Write the factored form using these numbers: ( x − 2 ) ( x − 9 ) .
The correct factorization is ( x − 9 ) ( x − 2 ) .
Explanation
Understanding the Problem We are given the quadratic expression x 2 − 11 x + 18 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 Right Numbers To factor the quadratic expression x 2 − 11 x + 18 , we need to find two numbers that multiply to 18 (the constant term) and add up to -11 (the coefficient of the x term). Let's call these two numbers a and b . We need to find a and b such that:
a × b = 18 a + b = − 11
Identifying the Correct Factors We can list the pairs of factors of 18:
1 and 18 2 and 9 3 and 6
Since we need the two numbers to add up to -11, we need to consider negative factors as well:
-1 and -18 -2 and -9 -3 and -6
We can see that -2 and -9 satisfy both conditions:
( − 2 ) × ( − 9 ) = 18 ( − 2 ) + ( − 9 ) = − 11
Writing the Factored Form Now that we have the two numbers, -2 and -9, we can write the factored form of the quadratic expression as:
( x − 2 ) ( x − 9 ) or ( x − 9 ) ( x − 2 )
Final Answer Therefore, the correct factorization of x 2 − 11 x + 18 is ( x − 2 ) ( x − 9 ) .
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. Imagine you are designing a rectangular garden and you know the area is represented by the expression x 2 − 11 x + 18 . By factoring this expression to ( x − 2 ) ( x − 9 ) , you determine the possible dimensions of the garden. If x represents a length, then the garden could have a width of ( x − 2 ) units and a length of ( x − 9 ) units. This helps in planning the layout and optimizing the use of space.
