"begin by factoring numerators/denominators"begin by factoring numerators/denominators
9x^2-4" is a "color(blue)"difference of squares"9x2−4 is a difference of squares
•color(white)(x)a^2-b^2=(a-b)(a+b)∙xa2−b2=(a−b)(a+b)
9x^2-4=(3x-2)(3x+2)9x2−4=(3x−2)(3x+2)
2x-2=2(x-1)larr" common factor of 2"2x−2=2(x−1)← common factor of 2
21x^2-2x-8larrcolor(blue)"factor using a-c method"21x2−2x−8←factor using a-c method
"the factors of the product "21xx-8=-168the factors of the product 21×−8=−168
"which sum to - 2 are - 14 and + 12"which sum to - 2 are - 14 and + 12
"split the middle term using these factors"split the middle term using these factors
21x^2-14x+12x-821x2−14x+12x−8
=7x(3x-2)+4(3x-2)=7x(3x−2)+4(3x−2)
=(3x-2)(7x+4)=(3x−2)(7x+4)
"the original can now be expressed as"the original can now be expressed as
((3x-2)(3x+2))/(2(x-1))-:((3x-2)(7x+4))/1(3x−2)(3x+2)2(x−1)÷(3x−2)(7x+4)1
"to divide the 2 fractions change division to multiply"to divide the 2 fractions change division to multiply
"and turn the second fraction upside down"and turn the second fraction upside down
"cancel common factors on numerator/denominator"cancel common factors on numerator/denominator
=(cancel((3x-2))(3x+2))/(2(x-1))xx1/(cancel((3x-2))(7x+4))
=(3x+2)/(2(x-1)(7x+4))
