1. **State the problem:** Solve the equation $$\sqrt{x}2x + 63 = x$$ for $x$. 2. **Rewrite the equation:** The equation is $$2x\sqrt{x} + 63 = x$$. 3. **Isolate the radical term:** Move $x$ to the left side: $$2x\sqrt{x} = x - 63$$ 4. **Express $\sqrt{x}$ as $x^{1/2}$:** $$2x \cdot x^{1/2} = x - 63$$ 5. **Combine powers of $x$:** $$2x^{3/2} = x - 63$$ 6. **Rewrite as:** $$2x^{3/2} - x + 63 = 0$$ 7. **Substitute $t = \sqrt{x} = x^{1/2}$, so $x = t^2$:** $$2(t^2)^{3/2} - t^2 + 63 = 0$$ Since $(t^2)^{3/2} = t^{3}$, the equation becomes: $$2t^3 - t^2 + 63 = 0$$ 8. **Solve the cubic equation:** $$2t^3 - t^2 + 63 = 0$$ 9. **Try rational roots:** Possible roots are factors of 63 over factors of 2, e.g., $\pm1, \pm3, \pm7, \pm9, \pm21, \pm63, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{7}{2}, \pm\frac{9}{2}, \pm\frac{21}{2}, \pm\frac{63}{2}$. 10. **Test $t = -3$:** $$2(-3)^3 - (-3)^2 + 63 = 2(-27) - 9 + 63 = -54 - 9 + 63 = 0$$ So $t = -3$ is a root. 11. **Factor out $(t + 3)$:** Divide $2t^3 - t^2 + 63$ by $(t + 3)$: $$\frac{2t^3 - t^2 + 0t + 63}{t + 3} = 2t^2 - 7t + 21$$ 12. **Solve quadratic $2t^2 - 7t + 21 = 0$:** Discriminant: $$\Delta = (-7)^2 - 4 \cdot 2 \cdot 21 = 49 - 168 = -119 < 0$$ No real roots. 13. **Real root is $t = -3$ only.** 14. **Recall $t = \sqrt{x} \geq 0$, but $t = -3$ is negative, so discard.** 15. **No real solutions for $x$.** **Final answer:** No real solution for $x$.