To simplify the expression 3/(2x) – 5(21/(8x² – 14x – 15)), we will start by factoring the quadratic expression in the denominator.
The expression 8x² – 14x – 15 can be factored. We need two numbers that multiply to 8 imes -15 = -120 and add to -14. These numbers are -20 and 6.
We can rewrite the quadratic:
8x² – 20x + 6x – 15
Now, we group and factor:
(8x² – 20x) + (6x – 15) = 4x(2x – 5) + 3(2x – 5) = (4x + 3)(2x – 5)
So, the simplified denominator becomes (4x + 3)(2x – 5).
Substituting back, we have:
3/(2x) – 5(21/((4x + 3)(2x – 5)))
To combine these fractions, we find a common denominator, which would be 2x(4x + 3)(2x – 5). Now, rewrite the first fraction:
3/(2x) = 3(4x + 3)(2x – 5)/(2x(4x + 3)(2x – 5))
Now both fractions have the same denominator:
[3(4x + 3)(2x – 5) – 5 imes 21]/(2x(4x + 3)(2x – 5))
Now calculate the numerator:
3(8x² – 10x + 6x – 15) – 105 = 3(8x² – 4x – 15) – 105
Expanding that:
24x² – 12x – 45 – 105 = 24x² – 12x – 150
So, the expression simplifies to:
(24x² – 12x – 150)/(2x(4x + 3)(2x – 5))
Lastly, you can further simplify the numerator if possible. Factor out common terms, and you will have the final simplified answer.