1. Stating the problem: Solve the equation $$7 - 7\sqrt{x} + 4 : 7\sqrt{3x} + 5 = 16,807$$. Note that the colon ":" likely represents division, so rewrite the equation as $$7 - 7\sqrt{x} + \frac{4}{7\sqrt{3x}} + 5 = 16,807$$. 2. Simplify the equation by combining like terms: $$7 + 5 = 12$$, so the equation becomes $$12 - 7\sqrt{x} + \frac{4}{7\sqrt{3x}} = 16,807$$. 3. Isolate the terms involving $x$: $$-7\sqrt{x} + \frac{4}{7\sqrt{3x}} = 16,807 - 12$$ which simplifies to $$-7\sqrt{x} + \frac{4}{7\sqrt{3x}} = 16,795$$. 4. Multiply both sides by $7\sqrt{3x}$ to eliminate the denominator: $$7\sqrt{3x} \times \left(-7\sqrt{x} + \frac{4}{7\sqrt{3x}}\right) = 16,795 \times 7\sqrt{3x}$$. 5. Distribute on the left side: $$7\sqrt{3x} \times (-7\sqrt{x}) + 7\sqrt{3x} \times \frac{4}{7\sqrt{3x}} = 16,795 \times 7\sqrt{3x}$$. 6. Simplify each term: - $$7\sqrt{3x} \times (-7\sqrt{x}) = -49 \sqrt{3x} \sqrt{x} = -49 \sqrt{3x^2} = -49 x \sqrt{3}$$. - $$7\sqrt{3x} \times \frac{4}{7\sqrt{3x}} = 4$$ (since the terms cancel). 7. So the equation becomes: $$-49 x \sqrt{3} + 4 = 16,795 \times 7 \sqrt{3x}$$. 8. Subtract 4 from both sides: $$-49 x \sqrt{3} = 16,795 \times 7 \sqrt{3x} - 4$$. 9. This is a complicated equation involving $x$ and $\sqrt{3x}$. To solve, let $y = \sqrt{x}$, so $x = y^2$ and $\sqrt{3x} = \sqrt{3} y$. 10. Substitute into the equation: $$-49 y^2 \sqrt{3} = 16,795 \times 7 \sqrt{3} y - 4$$. 11. Divide both sides by $\sqrt{3}$: $$-49 y^2 = 16,795 \times 7 y - \frac{4}{\sqrt{3}}$$. 12. Multiply both sides by $\cancel{1}$ (no cancellation here), rearranged as: $$-49 y^2 - 117,565 y + \frac{4}{\sqrt{3}} = 0$$. 13. Multiply entire equation by $\sqrt{3}$ to clear denominator: $$-49 y^2 \sqrt{3} - 117,565 y \sqrt{3} + 4 = 0$$. 14. This is a quadratic in $y$ with complicated coefficients. For simplicity, approximate $\sqrt{3} \approx 1.732$: $$-49 \times 1.732 y^2 - 117,565 \times 1.732 y + 4 = 0$$ $$-84.868 y^2 - 203,654.38 y + 4 = 0$$. 15. Multiply entire equation by $-1$ to get standard form: $$84.868 y^2 + 203,654.38 y - 4 = 0$$. 16. Use quadratic formula: $$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=84.868$, $b=203,654.38$, $c=-4$. 17. Calculate discriminant: $$b^2 - 4ac = (203,654.38)^2 - 4 \times 84.868 \times (-4) \approx 4.148 \times 10^{10} + 1,358.9 = 4.148 \times 10^{10}$$ (approximate). 18. Calculate $y$: $$y = \frac{-203,654.38 \pm \sqrt{4.148 \times 10^{10}}}{2 \times 84.868}$$ $$= \frac{-203,654.38 \pm 203,634.5}{169.736}$$. 19. Two solutions: - $$y_1 = \frac{-203,654.38 + 203,634.5}{169.736} = \frac{-19.88}{169.736} \approx -0.117$$ (discard negative since $y=\sqrt{x} \geq 0$). - $$y_2 = \frac{-203,654.38 - 203,634.5}{169.736} = \frac{-407,288.88}{169.736} \approx -2,400$$ (discard negative). 20. No positive solution for $y$, so no real solution for $x$. Final answer: No real solution for $x$ satisfies the equation.