1. Stating the problem: Solve the quadratic equation $$2x\sqrt{x} - 13x \sqrt{x} + 10 = 0$$. 2. Simplify the equation: Notice that the terms $2x\sqrt{x}$ and $-13x\sqrt{x}$ can be combined since they have the same variable part. 3. Combine like terms: $$2x\sqrt{x} - 13x\sqrt{x} = (2 - 13)x\sqrt{x} = -11x\sqrt{x}$$ 4. Rewrite the equation: $$-11x\sqrt{x} + 10 = 0$$ 5. Isolate the term with $x$: $$-11x\sqrt{x} = -10$$ 6. Divide both sides by $-11$: $$x\sqrt{x} = \frac{10}{11}$$ 7. Express $x\sqrt{x}$ as $x^{3/2}$: $$x^{3/2} = \frac{10}{11}$$ 8. Solve for $x$ by raising both sides to the power $\frac{2}{3}$: $$x = \left(\frac{10}{11}\right)^{\frac{2}{3}}$$ 9. This is the exact solution. For an approximate decimal value: $$x \approx 0.912$$ Final answer: $$x = \left(\frac{10}{11}\right)^{\frac{2}{3}} \approx 0.912$$