Answer:
D. X=6.08, x=-0.58
Step-by-step explanation:
Given the rational expression
[tex]\frac{2x}{x-4}-\frac{2x-5}{x^2-10x+24} = \frac{-3}{x-6}\\\frac{2x}{x-4}-\frac{2x-5}{x^2-4x-6x+24} = \frac{-3}{x-6}\\\frac{2x}{x-4}-\frac{2x-5}{x(x-4)-6(x-4)} = \frac{-3}{x-6}\\ \\\frac{2x}{x-4}-\frac{2x-5}{(x-4)(x-6)} = \frac{-3}{x-6}\\Rerrange;\\\frac{2x-5}{(x-4)(x-6)} = \frac{2x}{x-4}+ \frac{3}{x-6}\\\frac{2x-5}{(x-4)(x-6)} = \frac{2x(x-6)+3(x-4))}{(x-4)(x-6)}\\[/tex]
Cancel out the denominator on both sides
[tex]2x-5 = 2x(x-6)+3(x-4)\\Expand\\2x-5 =2x^2-12x+3x-12\\2x-5 = 2x^2-9x-12\\Equate \ to \ zero\\2x^2-9x-12-2x+5 = 0\\2x^2-11x - 7 = 0\\[/tex]
Factorize
[tex]x = -(-11)\pm\frac{\sqrt{(-11)^2-4(2)(-7)} }{2(2)}\\x = 11\pm\frac{\sqrt{121+56} }{4}\\x = 11\pm\frac{\sqrt{177} }{4}\\x = 11\pm\frac{13.3 }{4}\\x = \frac{11+13.3}{4} \ and \ x = \frac{11-13.3}{4} \\x = 6.08 \ and \ - 0.58[/tex]
Hence the required solution is X=6.08 and -0.58
