Question 84078
{{{((2x^2-x-3)/(3x^2-11x-20))*((3x^2+7x+4)/(4x^2+0x-9))}}} Start with the given expression


Factor the first numerator {{{2x^2-x-3}}}

*[invoke quadratic_factoring 2, -1,-3]

--------------------------------------------------------------------------------------------



Factor the first denominator {{{3x^2-11x-20}}}

*[invoke quadratic_factoring 3, -11,-20]

--------------------------------------------------------------------------------------------



Factor the second numerator {{{3x^2+7x+4}}}

*[invoke quadratic_factoring 3, 7,4]

--------------------------------------------------------------------------------------------



Factor the second denominator {{{4x^2+0x-9}}}

*[invoke quadratic_factoring 4, 0,-9]

--------------------------------------------------------------------------------------------





So after factoring everything, we get:


{{{(((1*x+1)(2*x+-3))/((3*x+4)(1*x+-5)))(((1*x+1)(3*x+4))/((2*x+3)(2*x+-3)))}}}    
 

{{{(((1*x+1) highlight((2*x+-3)))/(highlight((3*x+4)) (1*x+-5)))(((1*x+1) highlight((3*x+4)))/((2*x+3) highlight((2*x+-3))))}}} Notice we have these common terms




{{{(((1*x+1) cross((2*x+-3)))/(cross((3*x+4)) (1*x+-5)))(((1*x+1) cross((3*x+4)))/((2*x+3) cross((2*x+-3))))}}} They divide and cancel out



Leaving you with


{{{(((1*x+1))/( (1*x+-5)))(((1*x+1))/((2*x+3)))}}}

which is our simplified answer




Now if you're required, multiply the remaining terms


{{{((x + 1)(x + 1))/((x - 5)(2*x + 3))}}}

Final Answer (if you are required to multiply them back together):

{{{((x + 1)(x + 1))/((x - 5)(2*x + 3))=(x^2 + 2*x + 1)/(2*x^2 - 7*x - 15)}}}


So basically the original expression


{{{((2x^2-x-3)/(3x^2-11x-20))*((3x^2+7x+4)/(4x^2+0x-9))}}}


simplifies to this expression


{{{(x^2 + 2*x + 1)/(2*x^2 - 7*x - 15)}}}




Check:


Here's a way to check if you simplified it right. Simply graph the original expression

Original expression:


{{{((2x^2-x-3)/(3x^2-11x-20))*((3x^2+7x+4)/(4x^2+0x-9))}}}



Graph of {{{y=((2x^2-x-3)/(3x^2-11x-20))*((3x^2+7x+4)/(4x^2+0x-9))}}}:

{{{ graph( 300, 200, -6, 5, -10, 10, ((2x^2-x-3)/(3x^2-11x-20))*((3x^2+7x+4)/(4x^2+0x-9))) }}}


and graph the simplified expression


Simplified Expression:


{{{(x^2 + 2*x + 1)/(2*x^2 - 7*x - 15)}}}


Graph of {{{(x^2 + 2*x + 1)/(2*x^2 - 7*x - 15)}}}:

{{{ graph( 300, 200, -6, 5, -10, 10, (((1*x+1))/( (1*x+-5)))(((1*x+1))/((2*x+3)))) }}}


and you'll notice that the two equations plot the same curve (if they were on the same graph, they would overlap each other). Since this shows that the two equations are equivalent, this verifies our answer.