Solve Completing Square 3A165D

1. **State the problem:** Solve the quadratic equation $$3 + 2x^2 = -8x$$ by completing the square. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$2x^2 + 8x + 3 = 0$$ 3. **Divide through by the coefficient of $$x^2$$ to make it 1:** $$\frac{2x^2}{2} + \frac{8x}{2} + \frac{3}{2} = 0 \implies x^2 + 4x + \frac{3}{2} = 0$$ 4. **Isolate the constant term:** $$x^2 + 4x = -\frac{3}{2}$$ 5. **Complete the square:** Take half of the coefficient of $$x$$, which is $$4$$, half is $$2$$, then square it: $$2^2 = 4$$. Add and subtract $$4$$ on the left side: $$x^2 + 4x + 4 - 4 = -\frac{3}{2}$$ Rewrite as: $$\left(x + 2\right)^2 - 4 = -\frac{3}{2}$$ 6. **Move the subtracted square term to the right side:** $$\left(x + 2\right)^2 = -\frac{3}{2} + 4$$ Convert $$4$$ to fraction with denominator 2: $$4 = \frac{8}{2}$$ So: $$\left(x + 2\right)^2 = -\frac{3}{2} + \frac{8}{2} = \frac{5}{2}$$ 7. **Take the square root of both sides:** $$x + 2 = \pm \sqrt{\frac{5}{2}}$$ 8. **Simplify the square root:** $$\sqrt{\frac{5}{2}} = \frac{\sqrt{10}}{2}$$ So: $$x + 2 = \pm \frac{\sqrt{10}}{2}$$ 9. **Solve for $$x$$:** $$x = -2 \pm \frac{\sqrt{10}}{2}$$ 10. **Write the final answer:** $$x = -2 + \frac{\sqrt{10}}{2} \quad \text{or} \quad x = -2 - \frac{\sqrt{10}}{2}$$