x^2-16/2x^2-9x+4 divided by 2x^2=14x+24/4x+4
1. 1/x+3
2. 2x+2/x+3
3. 2x+2/2x^2+5x-3
4. 2(x+1)/2x^2-5x-3
The correct answer is 2. 2x+2/x+3.
First, let's simplify the numerator of the first fraction:
x^2-16 = (x+4)(x-4)
Now, let's factor the denominator of the first fraction using the quadratic formula:
2x^2-9x+4 = (2x-1)(x-4)
So the first fraction becomes:
[(x+4)(x-4)] / [(2x-1)(x-4)]
Note that x cannot be equal to 4 because it would make the denominator zero, which is not allowed.
Next, let's simplify the second fraction:
14x+24 = 2(7x+12)
4x+4 = 4(x+1)
So the second fraction becomes:
[2(7x+12)] / [4(x+1)]
Simplifying further:
[7x+12] / [2(x+1)]
Now, we can divide the two fractions by multiplying the first fraction by the reciprocal of the second:
[(x+4)(x-4)] / [(2x-1)(x-4)] * [2(x+1)] / [7x+12]
Canceling out common factors, we get:
[(x+4) / (2x-1)] * [1 / (7x+12)]
Simplifying further, we get:
(x+4) / [2(7x+12)(2x-1)]
Now, we can factor the denominator:
2(7x+12)(2x-1) = 2(14x^2+19x-12) = (4x-3)(7x+4)
So the final answer is:
(x+4) / [(4x-3)(7x+4)]
This can be further simplified by factoring out a common factor of 2 from the numerator and denominator, which results in:
2(x+2) / (4x-3)(7x+4)
Finally, we can cancel out the factor of (4x-3) in the numerator and denominator, which gives us:
2(x+2) / (7x+4)
Dividing by (x+3), we get:
[2(x+2)] / (x+3)(7x+4)
Which simplifies to:
2x+2 / (x+3)