simplify each expression

Question

X^2-4/x^2-9 divided by x+2/2x^2 +3x-9

Answer 1

(x²-4)/(x²-9) / (x+2)/(2x²+3x-9)

• x²-4 factors to (x+2)(x-2)
• x²-9 factors to (x+3)(x-3)
• 2x²+3x-9 factors to (2x-3)(x+3)  (use the "ac" method)

 

Substitute the factored terms into the original expression:

 

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

 

Now when you divide a number by a fraction, it's the same as multiplying by its reciprocal:

 

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

 

Cancel identical terms:

 

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

 

     (x-2)(2x-3)/(x+3) = (2x²-7x+6)/(x+3)

 

But note that x ≠ -3, -2, or 3 since these values would make the denominator of the original expression equal to zero, which is undefined.

 

 

Answer 2

x²-4         x+2

----- ÷ ------------

x2-9     2x²+3x-9

 

 

When dividing fractions, multiply the inverse of the fraction.

 

x²-4      2x²+3x-9

-----  •  -----------

x²-9         x+2

 

 

Factor each expression

 

(x+2)(x-2)(2x-3)(x+3)

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

    (x-3)(x+3)(x+2)

 

 

Notice that the factors (x+2) and (x+3) are in both the numerator and denominator

 

(x+2)(x-2)(2x-3)(x+3)
--------------------------
    (x-3)(x+3)(x+2)

 

 

So we are left with

(x-2)(2x-3)

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

   (x-3)

 

 

This is equivalent to

2x²-7x+6

-----------

     x-3