Question

solve this rational expression
(x^2 - 16 / 2x^2 -9x +4) / (2x^2 + 14x + 24 / 4x + 4)

Answer 1

what is the answer

Answer 2

To simplify this rational expression, we first need to factorise the numerator and denominator of each fraction.

For the numerator of the larger fraction:
x^2 - 16 can be factored as (x + 4)(x - 4)

For the denominator of the larger fraction:
2x^2 - 9x + 4 can be factored as (2x - 1)(x - 4)

So the larger fraction becomes:
((x + 4)(x - 4)) / ((2x - 1)(x - 4))

For the numerator of the smaller fraction:
2x^2 + 14x + 24 can be factored as 2(x + 4)(x + 3)

For the denominator of the smaller fraction:
4x + 4 can be factored as 4(x + 1)

So the smaller fraction becomes:
(2(x + 4)(x + 3)) / 4(x + 1)

Putting back these factored forms into the original expression:
((x + 4)(x - 4) / (2x - 1)(x - 4)) / (2(x + 4)(x + 3) / 4(x + 1))

To simplify further, we can multiply by the reciprocal of the denominator to get rid of the complex fraction:

((x + 4)(x - 4) / (2x - 1)(x - 4)) * (4(x + 1) / 2(x + 4)(x + 3))

Now we can cancel out the common terms from the numerator and the denominator to simplify the expression.

Answer 3

After canceling out common terms in the expression

((x + 4)(x - 4) / (2x - 1)(x - 4)) * (4(x + 1) / 2(x + 4)(x + 3)),

we are left with:

(4(x + 1) / (2x - 1)(x + 3)).

Therefore, the simplified form of the given rational expression is:

4(x + 1) / ((2x - 1)(x + 3))