Question

Factor completely:

A) 64x²-49y²
B) 20x²-45y²

Answer 1

$Use\\a^2-b^2=(a-b)(a+b)\\(ab)^n=a^nb^n\\(\sqrt{a})^2=a\ for\ a\geq0\\------------------------------\\A)\ 64x^2-49y^2=8^2x^2-7^2y^2=(8x)^2-(7y)^2=(8x-7y)(8x+7y)\\\\B)\ 20x^2-45y^2=(\sqrt{20})^2x^2-(\sqrt{45})^2y^2=(*)\\\\\sqrt{20}=\sqrt{4\cdot5}=\sqrt4\cdot\sqrt5=2\sqrt5\\\sqrt{45}=\sqrt{9\cdot5}=\sqrt9\cdot\sqrt5=3\sqrt5\\\\(*)=(2\sqrt5x)^2-(3\sqrt5y)^2=(2\sqrt5x-3\sqrt5y)(3\sqrt5x+3\sqrt5y)$

$or:\\\\20x^2-45y^2=5\cdot4x^2-5\cdot9y^2=5(4x^2-9y^2)=5[(2x)^2-(3y)^2]\\\\=5(2x-3y)(2x+3y)$

Answer 2

A) This is a difference of two squares. So your equation (a²x² - b²x²) where a and b can be square-rooted ends up being factored to (ax -by)(ax + by).

64x² - 49y² = (8x - 7y)(8x + 7y)

B) Factor out a GCF of 5 from both terms, then you have the same situation as equation A.

20x² - 45y² = 5(4x² - 9y²) = 5(2x - 3y)(2x + 3y)