What is $y^2-3 y-18$ in factored form?
Answer (2)
Find two numbers that multiply to -18 and add up to -3.
The numbers are 3 and -6.
Write the factored form using these numbers: ( y + 3 ) ( y − 6 ) .
The factored form of the quadratic is ( y + 3 ) ( y − 6 ) .
Explanation
Understanding the Problem We are given the quadratic expression y 2 − 3 y − 18 and we want to express it in factored form. Factoring a quadratic means rewriting it as a product of two binomials.
Finding the Right Numbers To factor the quadratic 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). Let's call these two numbers a and b . We need to find a and b such that:
a × b = − 18 a + b = − 3
Identifying the Correct Pair 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 adds up to -3:
3 + ( − 6 ) = − 3
So, the numbers we are looking for are 3 and -6.
Writing the Factored Form Now that we have the numbers 3 and -6, we can write the factored form of the quadratic expression as:
( y + 3 ) ( y − 6 )
Final Answer Therefore, the factored form of y 2 − 3 y − 18 is ( y + 3 ) ( y − 6 ) .
Examples
Factoring quadratics is useful in many real-world scenarios. For example, imagine you are designing a rectangular garden. You know the area of the garden needs to be represented by the equation y 2 − 3 y − 18 , where y is related to the dimensions of the garden. By factoring this quadratic into ( y + 3 ) ( y − 6 ) , you can determine the possible dimensions of the garden. This helps in planning the layout and optimizing the space.
The quadratic expression y 2 − 3 y − 18 can be factored into ( y + 3 ) ( y − 6 ) . This is accomplished by finding two numbers that multiply to − 18 and add up to − 3 . The numbers that satisfy this are 3 and − 6 .
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