Question

1+2а+1/а²-1 - а/а-1​

Answer 1

(1 + 2а + 1)/(а² - 1)  -  а/(а - 1) =

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

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

= 2/(а - 1) - а/(а - 1) = (2 - а)/(а - 1)

Answer 2

To simplify the expression, let's take it step by step:

1 + 2a + 1/(a² - 1) - a/(a - 1)

First, let's simplify the denominators:

a² - 1 = (a + 1)(a - 1) a - 1 = -(1 - a)

Now, we can rewrite the expression as:

1 + 2a + 1/[(a + 1)(a - 1)] - a/[-(1 - a)]

Next, we can simplify the expression by finding a common denominator for the fractions:

Denominator = (a + 1)(a - 1)(-(1 - a))

Now, let's rewrite each fraction with the common denominator:

(1 * (a + 1)(a - 1)(-(1 - a))) / [(a + 1)(a - 1)(-(1 - a))]

• (2a * (a + 1)(a - 1)(-(1 - a))) / [(a + 1)(a - 1)(-(1 - a))]
• (1/[(a + 1)(a - 1)(-(1 - a))])

• (a/[(a + 1)(a - 1)(-(1 - a))])

Combining the fractions with the common denominator:

(a + 1)(a - 1)(-(1 - a)) + 2a(a + 1)(a - 1)(-(1 - a)) + 1 - a

(a + 1)(a - 1)(-(1 - a))

Now, let's simplify the numerator:

(a + 1)(a - 1)(-(1 - a)) = -(a + 1)(a - 1)(a - 1 + 1) = -(a + 1)(a - 1)(a) = -a(a + 1)(a - 1)

Plugging in the simplified numerator:

-a(a + 1)(a - 1) + 2a(a + 1)(a - 1)(-(1 - a)) + 1 - a

(a + 1)(a - 1)(-(1 - a))

Next, let's simplify the denominator:

(a + 1)(a - 1)(-(1 - a)) = -(a + 1)(a - 1)(a - 1 + 1) = -(a + 1)(a - 1)(a) = -a(a + 1)(a - 1)

Plugging in the simplified denominator:

(-a(a + 1)(a - 1) + 2a(a + 1)(a - 1)(-(1 - a)) + 1 - a) / (-a(a + 1)(a - 1))

Now, we can cancel out common factors in the numerator and denominator:

-1 + 2(1 - a) + 1 - a

-(1 - a)

Simplifying the expression:

-1 + 2 - 2a + 1 - a

-1 + a

Combining like terms:

(-2a - a + 2 - 1 + 1) / (-1 + a)

(-2a)

Simplifying further:

(-3a) / (-1 + a)

Finally, we can simplify the expression to:

3a / (a - 1)