1. **State the problem:** Solve the equation $$\frac{52}{x+30} - \frac{52}{x} = \frac{27}{10}$$ for $x$. 2. **Formula and rules:** To solve equations involving fractions, find a common denominator to combine terms and then solve the resulting equation. 3. **Find common denominator and combine:** $$\frac{52}{x+30} - \frac{52}{x} = \frac{52x - 52(x+30)}{x(x+30)} = \frac{52x - 52x - 1560}{x(x+30)} = \frac{-1560}{x(x+30)}$$ 4. **Rewrite the equation:** $$\frac{-1560}{x(x+30)} = \frac{27}{10}$$ 5. **Cross multiply:** $$-1560 \times 10 = 27 \times x(x+30)$$ $$-15600 = 27x^2 + 810x$$ 6. **Bring all terms to one side:** $$27x^2 + 810x + 15600 = 0$$ 7. **Simplify by dividing all terms by 27:** $$\frac{27x^2}{\cancel{27}} + \frac{810x}{\cancel{27}} + \frac{15600}{\cancel{27}} = 0$$ $$x^2 + 30x + \frac{15600}{27} = 0$$ 8. **Simplify the constant term:** $$\frac{15600}{27} = \frac{5200}{9}$$ So the quadratic is: $$x^2 + 30x + \frac{5200}{9} = 0$$ 9. **Use quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=30$, $c=\frac{5200}{9}$. Calculate discriminant: $$\Delta = 30^2 - 4 \times 1 \times \frac{5200}{9} = 900 - \frac{20800}{9} = \frac{8100 - 20800}{9} = \frac{-12700}{9}$$ 10. **Since $\Delta < 0$, no real solutions exist.** **Final answer:** There are no real values of $x$ satisfying the equation.