What is $y^2-3 y-18$ in factored form?
Answer (2)
Find two numbers that multiply to -18 and add up 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 asked to factor the quadratic expression y 2 − 3 y − 18 . This means we need to find two binomials that, when multiplied together, give us the original quadratic expression.
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).
Listing Factor Pairs 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
Identifying the Correct Pair Now, let's check which of these pairs adds 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 adds up to -3.
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 ) .
Verification To verify our answer, 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 factored form is correct.
Final Answer Therefore, the factored form of y 2 − 3 y − 18 is ( y + 3 ) ( y − 6 ) .
Examples
Factoring quadratic expressions is useful in many real-world applications. For example, if you are designing a rectangular garden and you know the area can be represented by the expression y 2 − 3 y − 18 , where y is related to the dimensions, factoring the expression into ( y + 3 ) ( y − 6 ) helps you determine the possible lengths and widths of the garden. This can help in optimizing the layout and planning the space effectively. Factoring is also used in physics to solve projectile motion problems, where the height of an object is often described by a quadratic equation.
The quadratic expression y 2 − 3 y − 18 can be factored into ( y + 3 ) ( y − 6 ) . This is determined by finding two numbers that multiply to -18 and add to -3, which are 3 and -6. The verification by expanding the factors confirms that the original expression matches.
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