Question

question 2 of 25
which of the expressions below can be factored using the difference of squares method?

a. $25x^2 - 64y^2$

b. $17x^2 + 23y^2$

c. $25x^2 + 64y^2$

d. $17x^2 - 23y^2$

Explanation:

Step1: Recall difference of squares formula

The difference of squares formula is \(a^2 - b^2=(a + b)(a - b)\). For an expression to be factored using this method, it must be a difference (subtraction) of two perfect squares. A perfect square has a square root that is an integer or a monomial with integer coefficients.

Step2: Analyze Option A

For \(25x^{2}-64y^{2}\), we can rewrite \(25x^{2}=(5x)^{2}\) and \(64y^{2}=(8y)^{2}\). So it can be written as \((5x)^{2}-(8y)^{2}\), which fits the difference of squares form \(a^{2}-b^{2}\) where \(a = 5x\) and \(b=8y\).

Step3: Analyze Option B

The expression \(17x^{2}+23y^{2}\) is a sum (addition) of two terms, not a difference, so it cannot be factored using the difference of squares method.

Step4: Analyze Option C

The expression \(25x^{2}+64y^{2}\) is a sum (addition) of two terms, not a difference, so it cannot be factored using the difference of squares method.

Step5: Analyze Option D

For \(17x^{2}-23y^{2}\), \(17x^{2}=(\sqrt{17}x)^{2}\) and \(23y^{2}=(\sqrt{23}y)^{2}\), but \(\sqrt{17}\) and \(\sqrt{23}\) are not integers, so \(17x^{2}\) and \(23y^{2}\) are not perfect squares (in the context of integer - coefficient monomial squares for factoring with integer coefficients), so it does not fit the difference of squares form for factoring with integer coefficients.

Answer:

A. \(25x^{2}-64y^{2}\)