Problem 29 - Beginning and Intermediate Algebra

Question

Factor by grouping. $$9 x^{2}-13 x y+4 y^{2}$$

Step-by-Step Solution

Answer

The factored expression by grouping is \((x-y)(9x-4y)\).

Step 1: Split the middle term

First, we need to split the middle term, -13xy, into two terms that when added will give us -13xy and when multiplied will give us the product of the first and third terms (9x^2 and 4y^2). \(9x^{2}-13xy+4y^{2}\Rightarrow 9x^{2}-9xy-4xy+4y^{2}\) Notice that we have split -13xy into -9xy and -4xy.

Step 2: Group terms

Now, we will group the expression into pairs that have the GCF: \((9x^{2}-9xy)+(-4xy+4y^{2})\)

Step 3: Factor out the GCF from each group

Factor out the GCF from the first group and from the second group separately: 9x is the GCF of the first group, and 4y is the GCF of the second group: \(9x(x-y)+4y(-x+y)\) Notice that we have (x-y) and (-x+y) as factors in both groups. Since (-x+y) is the same as (x-y) multiplied by -1, we can rewrite our expression to have the same binomial in both terms: \(9x(x-y)-4y(x-y)\)

Step 4: Factor out the common binomial

Now, we can factor out the common binomial (x-y) from both terms. This leaves us with: \((x-y)(9x-4y)\) So, the factored form of the given expression is: \((x-y)(9x-4y)\)