What is $y^2-3 y-18$ in factored form?
Answer (2)
Find two numbers that multiply to -18 and add to -3.
Identify the numbers as 3 and -6.
Write the factored form as ( y + 3 ) ( y − 6 ) .
The factored form of the quadratic expression is ( y + 3 ) ( y − 6 ) .
Explanation
Understanding the Problem We are given the quadratic expression y 2 − 3 y − 18 and asked to factor it. Factoring involves expressing the quadratic as a product of two binomials.
Finding the Right Numbers To factor the quadratic expression y 2 − 3 y − 18 , we need to find two numbers that multiply to -18 (the constant term) and add up to -3 (the coefficient of the y term).
Identifying the Correct Pair Let's list the factor pairs of -18:
1 and -18 (sum is -17)
-1 and 18 (sum is 17)
2 and -9 (sum is -7)
-2 and 9 (sum is 7)
3 and -6 (sum is -3)
-3 and 6 (sum is 3)
We see that the pair 3 and -6 satisfy our conditions: 3 × − 6 = − 18 and 3 + ( − 6 ) = − 3 .
Writing the Factored Form Now we can write the factored form of the quadratic expression using the numbers we found:
y 2 − 3 y − 18 = ( y + 3 ) ( y − 6 )
Verification To verify our factoring, we can expand the factored form:
( y + 3 ) ( y − 6 ) = y 2 − 6 y + 3 y − 18 = y 2 − 3 y − 18
This matches the original quadratic expression, so our factoring is correct.
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, suppose you want to design a rectangular garden with an area represented by the expression y 2 − 3 y − 18 square feet. By factoring this expression into ( y + 3 ) ( y − 6 ) , you determine the possible dimensions of the garden. If y = 10 , then the dimensions would be ( 10 + 3 ) = 13 feet and ( 10 − 6 ) = 4 feet, giving an area of 52 square feet. Factoring helps in optimizing designs and solving area-related problems.
The quadratic expression y 2 − 3 y − 18 can be factored into ( y + 3 ) ( y − 6 ) by finding two numbers that multiply to -18 and add to -3. The numbers are 3 and -6. This factorization helps in simplifying algebraic expressions and solving equations.
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