# simplify a^2-1/a^2+2a-3

mathematicianmckenna | High School Teacher | eNotes Newbie

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Step 1

Factor the numerator using the difference of two squares:

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

Step 2

Factor the denominator. Here were are looking for values that ADD to = -3 and MULTIPLY to = +2. We can use  +3 and -1.

So,

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

Step 3

Rewrite the orginal fraction with the factored numerator and denominator

(a+1)(a-1) / (a+3)(a-1)

Step 4

Cancel out the like terms (a-1) , we are left with

(a+1) / (a+3)

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

First, you need to put the numerator and denominator within brackets, such as:

`((a^2 - 1))/((a^2 + 2a - 3))`

We notice that the numerator is a difference of two squares that returns the special product:

`a^2 - 1 = (a-1)(a+1)`

We'll decompose the denominator in it's factors. For this reason, we'll apply the quadratic formula, to determine the roots of the expression from denominator.

a1 = (-2+`sqrt(4 + 12)` )/2

a1 = (-2+4)/2

a1 = 1

a2 = (-2-4)/2

a2 = -3

The denominator could be written as a product of linear factors:

`a^2 + 2a - 3 = (a - 1)(a + 3)`

We'll re-write the given expression:

`((a^2 - 1))/((a^2 + 2a - 3))` = `[(a-1)(a+1)]/[(a-1)(a+3)]`

We'll reduce the fraction and we'll get:

`((a^2 - 1))/((a^2 + 2a - 3)) = ((a + 1))/((a+3))`