To simplify the expression 3/(2x – 5) – 21/(8x2 – 14x – 15), we first need to factor the quadratic in the denominator of the second fraction.
1. **Factor the denominator**: The expression 8x2 – 14x – 15 can be factored. To do this, we look for two numbers that multiply to 8 imes -15 = -120 and add to -14. These numbers are -20 and 6. So, we rewrite -14x as -20x + 6x.
– The expression becomes 8x2 – 20x + 6x – 15.
– Next, we factor by grouping:
8x(x – 2.5) + 6(x – 2.5)
– Which factors to (x – 2.5)(8x + 6). Finally, we can write 8x + 6 as 2(4x + 3). So, our factored expression becomes:
8x2 – 14x – 15 = (4x + 3)(2x – 5)
2. **Rewrite the original expression**: Now, substitute the factored form back into the original expression:
3/(2x – 5) – 21/((4x + 3)(2x – 5))
3. **Combine the fractions**: To subtract the two fractions, we need a common denominator:
– The common denominator is (2x – 5)(4x + 3).
– Rewrite the first fraction with the common denominator:
3/(2x – 5) imes (4x + 3)/(4x + 3) = 3(4x + 3)/((2x – 5)(4x + 3))
– Now, the expression looks like:
(3(4x + 3) – 21)/((2x – 5)(4x + 3))
4. **Simplify the numerator**:
– Expand 3(4x + 3):
12x + 9 – 21
– Which simplifies to 12x – 12.
5. **Write the final expression**:
– So our expression is finally:
(12x – 12)/((2x – 5)(4x + 3))
– We can factor out 12 from the numerator:
12(x – 1)/((2x – 5)(4x + 3))
Thus, the simplified form of the expression 3/(2x – 5) – 21/(8x2 – 14x – 15) is:
12(x – 1)/((2x – 5)(4x + 3)).