1. **Problem statement:** Three charges lie along the x-axis: $q_2=6.0$ nC at $x=0$, $q_1=15.0$ nC at $x=2.00$ m, and $q_3$ at an unknown position $x$ between them. The net force on $q_3$ is zero. Find the coordinate $x$ of $q_3$. 2. **Formula:** The electrostatic force between two point charges is given by Coulomb's law: $$F = k \frac{|q_a q_b|}{r^2}$$ where $k=8.99 \times 10^9$ N m$^2$/C$^2$, $q_a$ and $q_b$ are charges, and $r$ is the distance between them. 3. **Setup forces on $q_3$:** - Force from $q_2$ on $q_3$ is $F_{23} = k \frac{q_2 q_3}{x^2}$ directed leftwards. - Force from $q_1$ on $q_3$ is $F_{13} = k \frac{q_1 q_3}{(2.00 - x)^2}$ directed rightwards. 4. Since net force on $q_3$ is zero, forces balance: $$F_{23} = F_{13}$$ $$k \frac{q_2 q_3}{x^2} = k \frac{q_1 q_3}{(2.00 - x)^2}$$ 5. Cancel common factors $k$ and $q_3$: $$\cancel{k} \frac{q_2 \cancel{q_3}}{x^2} = \cancel{k} \frac{q_1 \cancel{q_3}}{(2.00 - x)^2}$$ $$\frac{q_2}{x^2} = \frac{q_1}{(2.00 - x)^2}$$ 6. Cross multiply: $$q_2 (2.00 - x)^2 = q_1 x^2$$ 7. Substitute values $q_1=15.0$, $q_2=6.0$ (in nC, units cancel): $$6.0 (2.00 - x)^2 = 15.0 x^2$$ 8. Divide both sides by 3 to simplify: $$\frac{6.0}{3} (2.00 - x)^2 = \frac{15.0}{3} x^2$$ $$2 (2.00 - x)^2 = 5 x^2$$ 9. Expand $(2.00 - x)^2$: $$2 (4.00 - 4.00 x + x^2) = 5 x^2$$ $$8.00 - 8.00 x + 2 x^2 = 5 x^2$$ 10. Rearrange terms: $$8.00 - 8.00 x + 2 x^2 - 5 x^2 = 0$$ $$8.00 - 8.00 x - 3 x^2 = 0$$ 11. Multiply entire equation by $-1$ for standard form: $$3 x^2 + 8.00 x - 8.00 = 0$$ 12. Solve quadratic equation $3 x^2 + 8 x - 8 = 0$ using quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a}$$ where $a=3$, $b=8$, $c=-8$. 13. Calculate discriminant: $$\Delta = 8^2 - 4 \times 3 \times (-8) = 64 + 96 = 160$$ 14. Calculate roots: $$x = \frac{-8 \pm \sqrt{160}}{6} = \frac{-8 \pm 4 \sqrt{10}}{6}$$ 15. Approximate values: $$x_1 = \frac{-8 + 12.65}{6} = \frac{4.65}{6} \approx 0.775$$ $$x_2 = \frac{-8 - 12.65}{6} = \frac{-20.65}{6} \approx -3.44$$ 16. Since $q_3$ lies between $0$ and $2.00$ m, discard negative root. **Final answer:** $$\boxed{x \approx 0.775 \text{ meters}}$$