How do you multiply #(6x^2 + x - 12 )/(2x^2 - 5x - 12 )*(3x^2 - 14x + 8)/( 9x^2 - 18x + 8)#?

Question

Abigail Allen · Accepted Answer

It may help to factor each of the quadratics into linear factors first - There may be some terms we can eliminate :-) #6x^2+x-12 = (3x-4)(2x+3)#
#3x^2-14x+8 = (3x-2)(x-4)#
#2x^2-5x-12 = (2x+3)(x-4)#
#9x^2-18x+8 = (3x-4)(3x-2)# So #(6x^2+x-12)/(2x^2-5x-12)*(3x^2-14x+8)/(9x^2-18x+8)# #=((3x-4)(2x+3))/((2x+3)(x-4))*((3x-2)(x-4))/((3x-4)(3x-2))# #=(3x-4)/(x-4)*(x-4)/(3x-4)# #=1# Note that this equation is only valid when the denominator is non-zero, which will be when #x# is not one of the values:
#-3/2#, #4#, #4/3# or #2/3#.

Eva Adamson · Answer

You can factorize all these quadratic equations first.

In order to factorize them, all tou need to do is find its roots and then turn each root into a factor by equaling it to zero.

For example, let's say you've found #x=1/2#. Then, your factor will be #(2x-1)=0#.

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

Now, we can proceed to cancel some binomials that are both dividing and multiplying this equation.

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

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

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

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

Thus, your result is #1#. :)

Avery Alexander · Answer

undefined To multiply the given expressions, you need to multiply the numerators together and multiply the denominators together. Then simplify the resulting expression if possible.

The numerator of the product will be (6x^2 + x - 12) * (3x^2 - 14x + 8).

The denominator of the product will be (2x^2 - 5x - 12) * (9x^2 - 18x + 8).

Multiplying the numerators and denominators, we get:

Numerator: (6x^2 + x - 12) * (3x^2 - 14x + 8) = 18x^4 - 84x^3 + 48x^2 + 3x^3 - 14x^2 + 8x - 12x^2 + 56x - 32 = 18x^4 - 81x^3 + 22x^2 + 64x - 32.

Denominator: (2x^2 - 5x - 12) * (9x^2 - 18x + 8) = 18x^4 - 81x^3 + 22x^2 + 64x - 32.

Therefore, the simplified expression is (18x^4 - 81x^3 + 22x^2 + 64x - 32) / (18x^4 - 81x^3 + 22x^2 + 64x - 32).