b) [tex]x^2-9 x+18[/tex]
Answer (1)
Find two numbers that multiply to 18 and add up to -9: -3 and -6.
Express the quadratic in factored form using these numbers: ( x − 3 ) ( x − 6 ) .
Verify the factored form by expanding it to ensure it matches the original quadratic expression.
The factored form of x 2 − 9 x + 18 is ( x − 3 ) ( x − 6 ) .
Explanation
Understanding the Problem We are given the quadratic expression x 2 − 9 x + 18 and our goal is to factor it. Factoring a quadratic means expressing it as a product of two binomials.
Finding the Factors To factor the quadratic x 2 − 9 x + 18 , we need to find two numbers that multiply to 18 (the constant term) and add up to -9 (the coefficient of the x term). Let's think of factors of 18:
1 and 18 2 and 9 3 and 6
Since we need the numbers to add up to -9, we should consider the negative versions of these factors. The pair -3 and -6 satisfy both conditions: ( − 3 ) × ( − 6 ) = 18 and ( − 3 ) + ( − 6 ) = − 9 .
Writing the Factored Form Now that we have the two numbers, -3 and -6, we can write the quadratic expression in factored form as ( x − 3 ) ( x − 6 ) .
Checking the Answer To check our answer, we can expand the factored form:
( x − 3 ) ( x − 6 ) = x ( x − 6 ) − 3 ( x − 6 ) = x 2 − 6 x − 3 x + 18 = x 2 − 9 x + 18 . This matches the original quadratic expression, so our factoring is correct.
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to analyze the stability of structures, economists use it to model supply and demand curves, and computer scientists use it to optimize algorithms. Suppose you want to build a rectangular garden with an area represented by the quadratic expression x 2 − 9 x + 18 . By factoring this expression into ( x − 3 ) ( x − 6 ) , you determine that the dimensions of the garden can be ( x − 3 ) and ( x − 6 ) . If you want the garden to have an area of 0, you can solve for x by setting each factor to zero, giving you possible values for the dimensions of the garden.
