Factor.
$y^2+7 y-18$
Answer (1)
Find two numbers that multiply to -18 and add up to 7.
The numbers are 9 and -2.
Write the factored form using these numbers: ( y + 9 ) ( y − 2 ) .
The factored form of the quadratic expression is ( y + 9 ) ( y − 2 ) .
Explanation
Understanding the Problem We are given the quadratic expression y 2 + 7 y − 18 and our goal is to factor it into the form ( y + a ) ( y + b ) , where a and b are numbers such that a e q b .
Finding the Right Numbers To factor the quadratic expression y 2 + 7 y − 18 , we need to find two numbers that multiply to -18 (the constant term) and add up to 7 (the coefficient of the y term). Let's call these two numbers a and b . So, we need to find a and b such that:
a × b = − 18 a + b = 7
Identifying the Correct Pair Let's list the pairs of factors 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 pair adds up to 7:
1 + ( − 18 ) = − 17 − 1 + 18 = 17 2 + ( − 9 ) = − 7 − 2 + 9 = 7 3 + ( − 6 ) = − 3 − 3 + 6 = 3
The pair -2 and 9 satisfies the conditions.
Writing the Factored Form Since we found that -2 and 9 are the numbers we need, we can write the factored form of the quadratic expression as:
( y − 2 ) ( y + 9 )
Final Answer Therefore, the factored form of the quadratic expression y 2 + 7 y − 18 is ( y + 9 ) ( y − 2 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For instance, imagine you are designing a rectangular garden and you know the area can be represented by the expression y 2 + 7 y − 18 , where y is a variable related to the dimensions. By factoring this expression into ( y + 9 ) ( y − 2 ) , you determine the possible lengths and widths of the garden. This helps you plan the layout and optimize the use of space, ensuring the garden fits your desired specifications.
