d) [tex]x^2+6 x-27[/tex]
Answer (1)
Find two numbers that multiply to -27 and add to 6: -3 and 9.
Rewrite the quadratic: x 2 − 3 x + 9 x − 27 .
Factor by grouping: x ( x − 3 ) + 9 ( x − 3 ) .
The factored form is ( x − 3 ) ( x + 9 ) .
Explanation
Understanding the Problem We are given the quadratic expression x 2 + 6 x − 27 and our goal is to factor it. Factoring a quadratic expression means rewriting it as a product of two binomials.
Finding the Right Numbers To factor the quadratic expression x 2 + 6 x − 27 , we need to find two numbers that multiply to -27 (the constant term) and add up to 6 (the coefficient of the x term). Let's call these two numbers a and b . So, we need to find a and b such that:
a × b = − 27 a + b = 6
Identifying the Correct Pair of Factors Let's list the pairs of factors of -27:
1 and -27 -1 and 27 3 and -9 -3 and 9
Now, let's check which pair adds up to 6:
1 + ( − 27 ) = − 26 − 1 + 27 = 26 3 + ( − 9 ) = − 6 − 3 + 9 = 6
So, the numbers we are looking for are -3 and 9.
Rewriting the Middle Term Now we can rewrite the middle term of the quadratic expression using these two numbers:
x 2 + 6 x − 27 = x 2 − 3 x + 9 x − 27
Factoring by Grouping Next, we factor by grouping:
x 2 − 3 x + 9 x − 27 = x ( x − 3 ) + 9 ( x − 3 )
Factoring out the Common Factor Now we can factor out the common binomial factor ( x − 3 ) :
x ( x − 3 ) + 9 ( x − 3 ) = ( x − 3 ) ( x + 9 )
Final Factored Form So, the factored form of the quadratic expression x 2 + 6 x − 27 is ( x − 3 ) ( x + 9 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, if you are designing a rectangular garden and you know the area can be represented by the quadratic expression x 2 + 6 x − 27 , factoring it into ( x − 3 ) ( x + 9 ) helps you determine the possible dimensions (length and width) of the garden in terms of x . This allows you to plan the layout and optimize the use of space based on the value of x .
