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Chapter 10: Problem 24

Factor. $$\left(x^{2}-4 x+4\right)-y^{2}$$

Short Answer

Expert verified
The factored form of the expression \(\left(x^{2}-4 x+4\right)-y^{2}\) is \((x-2+y)(x-2-y)\).

Step by step solution

01

Identify the Square Terms

Start the exercise by identifying the squares in the expression so as to decide how you will factor. In the given expression, \(\left(x^{2}-4 x+4\right)-y^{2}\), the two squares are \((x-2)^2\) and \(y^2\). This implies that your a and b values are \((x-2)\) and \(y\) respectively.
02

Apply the Difference of Squares Formula

With a as \((x-2)\) and b as \(y\), apply the difference of squares formula, \(a^2 - b^2 = (a+b)(a-b)\). Substituting your a and b gives the factors as: \((x-2+y)(x-2-y)\).
03

Confirmation

Confirm the result by expanding the factors to ensure that they give the original expression. If \((x-2+y)(x-2-y)\) is expanded, we indeed get the original expression \(\left(x^{2}-4 x+4\right)-y^{2}\).

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