Answer:
A. x = 6.08, x = -0.58
Step-by-step explanation:
- [tex]\displaystyle \frac{2x}{x-4}-\frac{2x-5}{x^2-10x+24}=\frac{-3}{x-6}[/tex]
1. Factor the denominator.
- [tex]\displaystyle \frac{2x}{x-4}-\frac{2x-5}{(x-6)(x-4)}=\frac{-3}{x-6}[/tex]
2. Multiply the leftmost fraction by (x-6)/(x-6) to get common denominators.
- [tex]\displaystyle \Big ( \frac{x-6}{x-6} \Big ) \frac{2x}{x-4}-\frac{2x-5}{(x-6)(x-4)}=\frac{-3}{x-6}[/tex]
3. Simplify.
- [tex]\displaystyle \frac{2x^2-12x}{(x-4)(x-6)}-\frac{2x-5}{(x-6)(x-4)}=\frac{-3}{x-6}[/tex]
4. Combine like terms.
- [tex]\displaystyle \frac{2x^2-14x+5}{(x-4)(x-6)}=\frac{-3}{x-6}[/tex]
5. Multiply the right side of the equation by (x-4)/(x-4).
- [tex]\displaystyle \frac{2x^2-14x+5}{(x-4)(x-6)}=\frac{-3}{x-6} \Big ( \frac{x-4}{x-4} \Big )[/tex]
6. Simplify.
- [tex]\displaystyle \frac{2x^2-14x+5}{(x-4)(x-6)}=\frac{-3x+12}{(x-6)(x-4)}[/tex]
7. "Cancel" out the denominators because they are equivalent.
- [tex]2x^2-14x+5=-3x+12[/tex]
8. Set the equation equal to 0.
Quadratic formula:
- [tex]\displaystyle x = \frac{-b\pm \sqrt{b^2-4ac} }{2a}[/tex]
9. Factor using the quadratic formula.
- [tex]\displaystyle x = \frac{-(-11)\pm \sqrt{(-11)^2-4(2)(-7)} }{2(2)}[/tex]
10. Multiply and simplify.
- [tex]\displaystyle x = \frac{11 \pm \sqrt{177} }{4}[/tex]
11. Plug this into your calculator and solve for x.
- [tex]x=6.07603367 \approx 6.08[/tex]
- [tex]x=-0.57603367 \approx -0.58[/tex]
The correct answer is A. x = 6.08, x = -0.58.
