How do you simplify #(x^2 - 16 )/( 2x^2 - 9x + 4) div (2x^2 + 14x + 24) /( 4x + 4)#?

Question

Ethan Adkins · Accepted Answer

#(2(x+1))/((2x-1)(x+3))#

By factoring note that:

#(x^2-16)/(2x^2-9x+4)÷(2x^2+14x+24)/(4x+4)=#

#((x+4)(x-4))/((2x-1)(x-4))*(4(x+1))/(2(x+3)(x+4))#

Cancel like terms:

#(cancel((x+4))cancel((x-4)))/((2x-1)cancel((x-4)))*(cancel(4)color(blue)(2)(x+1))/(cancel(2) (x+3)cancel((x+4)))=#

#(2(x+1))/((2x-1)(x+3))#

Jameson Allen · Answer

undefined To simplify the expression, we can start by factoring the numerator and denominator of the first fraction:

(x^2 - 16) = (x - 4)(x + 4)
(2x^2 - 9x + 4) = (2x - 1)(x - 4)

Next, we can factor the numerator and denominator of the second fraction:

(2x^2 + 14x + 24) = 2(x^2 + 7x + 12) = 2(x + 3)(x + 4)
(4x + 4) = 4(x + 1)

Now, we can rewrite the expression as a division of fractions:

[(x - 4)(x + 4)] / [(2x - 1)(x - 4)] ÷ [(2(x + 3)(x + 4)) / (4(x + 1))]

To divide fractions, we can multiply the first fraction by the reciprocal of the second fraction:

[(x - 4)(x + 4)] / [(2x - 1)(x - 4)] * [(4(x + 1)) / (2(x + 3)(x + 4))]

Next, we can cancel out common factors:

[(x - 4)(x + 4)] / [(2x - 1)(x - 4)] * [(4(x + 1)) / (2(x + 3)(x + 4))]
= [(x - 4) / (2x - 1)] * [(4(x + 1)) / (2(x + 3))]

Finally, we can simplify further by canceling out common factors:

[(x - 4) / (2x - 1)] * [(4(x + 1)) / (2(x + 3))]
= [(x - 4)(x + 1)] / [(2x - 1)(x + 3)]