What should be subtracted from x² – xy + y² to make it a perfect square?
A. xy
B. -xy
C. Both (xy) & (-xy)
D. None of these
Explanation and Detailed Step-by-Step Solution:
The problem asks what needs to be subtracted from the quadratic expression x² – xy + y² to make it a perfect square. Let’s go through the solution step-by-step.
Step 1: Understanding the Expression
The given expression is x² – xy + y². We need to determine what, when subtracted from this expression, will turn it into a perfect square trinomial.
Step 2: Identifying the Perfect Square
A perfect square trinomial is of the form (a – b)² = a² – 2ab + b² , which means there are specific values of a and b that can transform the expression into this format.
Looking at the form of x² – xy + y², it closely resembles x² – 2xy + y², which is the expanded form of (x – y)².
Step 3: What Needs to be Subtracted?
In the expression x² – xy + y², the middle term is -xy, but in a perfect square trinomial like (x – y)², the middle term should be -2xy. This means we are short by another -xy to complete the perfect square.
To turn x² – xy + y² into a perfect square, we must subtract an additional -xy to make the middle term -2xy.
Step 4: Subtracting -xy
Now, subtract -xy from the original expression:
x² – xy + y² – (-xy) = x² – 2xy + y²
This simplifies to:
(x – y)²
which is a perfect square.
Conclusion:
The correct answer is B. -xy. Subtracting -xy from the expression x² – xy + y² results in (x – y)², which is a perfect square.
Key Takeaways:
- A perfect square trinomial takes the form a² – 2ab + b².
- To make x² – xy + y² a perfect square, subtract -xy, which transforms the expression into (x – y)².
- This is an important technique when dealing with quadratic expressions and simplifying them to find perfect squares.
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