Factor.
$y^2-3 y-18$
Answer (1)
Find two numbers that multiply to -18 and add to -3: 3 and -6.
Write the factored form using these numbers: ( y + 3 ) ( y − 6 ) .
Verify the factorization by expanding: ( y + 3 ) ( y − 6 ) = y 2 − 3 y − 18 .
The factored form of the quadratic expression is ( y + 3 ) ( y − 6 ) .
Explanation
Understanding the Problem We are asked to factor the quadratic expression y 2 − 3 y − 18 . This means we want to find two binomials that, when multiplied together, give us the original quadratic expression.
Finding the Correct Factors To factor the quadratic expression y 2 − 3 y − 18 , we need to find two numbers that multiply to -18 and add to -3. Let's list the factor pairs of -18:
1 and -18
-1 and 18
2 and -9
-2 and 9
3 and -6
-3 and 6
Now, let's check which of these pairs add up to -3:
1 + (-18) = -17
-1 + 18 = 17
2 + (-9) = -7
-2 + 9 = 7
3 + (-6) = -3
-3 + 6 = 3
The pair 3 and -6 satisfy the condition since 3 + ( − 6 ) = − 3 and 3 × ( − 6 ) = − 18 .
Writing the Factored Form Now that we have found the two numbers, 3 and -6, we can write the factored form of the quadratic expression as ( y + 3 ) ( y − 6 ) .
Verification To verify our factorization, we can expand ( y + 3 ) ( y − 6 ) :
( y + 3 ) ( y − 6 ) = y ( y − 6 ) + 3 ( y − 6 ) = y 2 − 6 y + 3 y − 18 = y 2 − 3 y − 18 This matches the original quadratic expression, so our factorization is correct.
Final Answer The factored form of the quadratic expression y 2 − 3 y − 18 is ( y + 3 ) ( y − 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 analyze the stability of structures, economists use it to model supply and demand curves, and computer scientists use it to optimize algorithms. Imagine you are designing a rectangular garden with an area represented by the expression y 2 − 3 y − 18 . By factoring this expression into ( y + 3 ) ( y − 6 ) , you determine the possible dimensions of the garden. This allows you to plan the layout efficiently, ensuring that the garden meets specific area requirements while optimizing space and resources.
