Final answer:
To factor the expression x²-7x+7y-y² completely, first, rewrite it by grouping, then factor by recognizing the difference of squares and common factors. The final factored form of the expression is (x - y)(x + y - 7).
Step-by-step explanation:
The question asks to factor completely the given quadratic expression x²-7x+7y-y².
First, we need to recognize any patterns or possible groupings. In this case, we can rewrite the given expression by grouping terms:
x² - 7x + 7y - y² = (x² - y²) - (7x - 7y)
Next, we notice that x² - y² is a difference of squares, which can be factored as (x + y)(x - y). The second part, 7x - 7y, can be factored by taking out the common factor of 7, giving us 7(x - y).
After factoring, we get:
(x + y)(x - y) - 7(x - y)
From here, we can factor out the common (x - y) from both terms, leading to the completely factored form:
(x - y)(x + y - 7)
This expression cannot be factored further, so this is our final answer.