Simplify the expresions and solve the equations

Question

1. z² - 3z+2 over z² + 3z-4 multiplied by z² + 6z+8 over z² -4 all of this equals to z

 

2. 2h² -2h-4 over (h+1)(h-2) plus h² + h-2 over (h+2)(h+1)

Answer 1

 

1. (z²-3z+2) / (z²+3z-4)  *  (z²+6z+8) / (z²-4) = z

 

Factor the left hand side:

 

    (z-2)(z-1) / ( (z+4)(z-1) )  *  (z+4)(z+2) / ( (z-2)(z+2) ) = z

 

Cancel out like terms:

 

    (z-2)/(z+4) * (z+4)/(z-2) = z

 

    (z-2)(z+4) / ((z+4)(z-2) ) = z

 

Cancel like terms again:

 

    1 = z

 

So we find that z = 1. Now we need to check and make sure that when we plug this back into the equality, we do not divide by zero:

 

 (1²-3*1+2)/(1²+3*1-4)  *  (1²+6*1+8)/(1²-4) = 1

 

 

 0/0 * 15/-3 = 1

 

So when we plug z = 1 into the original equation, one of our terms turns out to be 0/0. This is impossible, so there is no solution.

 

2. (2h²-2h-4)/( (h+1)(h-2) ) + (h²+h-2)/( (h+2)(h+1) )

 

Factor a two out of the numerator in the first term:

 

    2*(h²-h-2)/( (h+1)(h-2) ) + (h²+h-2)/( (h+2)(h+1) )

 

Factor out our polynomials:

 

    2*(h-2)(h+1)/( (h+1)(h-2) ) + (h+2)(h-1)/( (h+2)(h+1) )

 

Cancel our like terms:

 

   2*(h-2)/(h-2) + (h-1)/(h+1)

 

Get a common denominator:

 

   2*(h-2)(h+1)/( (h-2)(h+1) ) + (h-1)(h-2)/( (h-2)(h+1) )

 

Add

 

   (2*(h-2)(h+1) + (h-1)(h-2) )/( (h-2)(h+1) )

 

Multiply out:

 

  (2(h²-h-2) + h²-3h+2)/( (h-2)(h+1) )

 

 =(2h²-2h-4+h²-3h+2)/( (h-2)(h+1) )

 

 =(3h²-5h-2)/( (h-2)(h+1) )

 

factor the numerator:

 

  = (3h+1)(h-2)/( (h-2)(h+1) )

 

  = (3h+1)/(h+1)

 

Answer 2

Could you try to write the expressions a little clearer? That's difficult to try to read.

Answer 3

z^2-3z+2= (z-1)(z-2)

 

z^2+3z-4= (z-1)(z+4)

z^2+6z+8=(z+2)(z+4)

 

z^2-4=(z-2)(z+2)

 

Putting all this together as per question----

 

((z-1)(z-2)/(z-1)(z+4))*((z+2)(z+4)/(z-2)(z+2))

 

=1=z

 

z=1

 

Second part---------

 

2h^2-2h-4=2(h-2)(h+1)

 

h^2+h-2= (h+2)(h-1)

 

Putting all this together as per question------

 

2(h-2)(h+1)/(h+1)(h-2)=2

 

2+((h-1)(h+1))

 

(3h+1)/(h+1)