Step 1: Divide the equation by the coefficient of $x^2$, which is $2$. $$2x^2 + 4x + 1 = 0$$ $$\frac{2x^2}{2} + \frac{4x}{2} + \frac{1}{2} = \frac{0}{2}$$ $$x^2 + 2x + \frac{1}{2} = 0$$ Step 2: Move the constant term to the right side of the equation. $$x^2 + 2x = -\frac{1}{2}$$ Step 3: Complete the square on the left side. To do this, take half of the coefficient of $x$ (which is $2$), square it, and add it to both sides of the equation. Half of the coefficient of $x$ is $\frac{2}{2} = 1$. Squaring it gives $1^2 = 1$. $$x^2 + 2x + 1 = -\frac{1}{2} + 1$$ Step 4: Factor the left side as a perfect square and simplify the right side. $$(x+1)^2 = \frac{1}{2}$$ Step 5: Take the square root of both sides. $$\sqrt{(x+1)^2} = \pm\sqrt{\frac{1}{2}}$$ $$x+1 = \pm\frac{1}{\sqrt{2}}$$ To rationalize the denominator, multiply the numerator and denominator by $\sqrt{2}$: $$x+1 = \pm\frac{\sqrt{2}}{2}$$ Step 6: Solve for $x$. $$x = -1 \pm \frac{\sqrt{2}}{2}$$ The two solutions are: $$x_1 = -1 + \frac{\sqrt{2}}{2}$$ $$x_2 = -1 - \frac{\sqrt{2}}{2}$$ The solutions can also be written as: $$x = \frac{-2 \pm \sqrt{2}}{2}$$ The final answers are $\boxed{x = -1 + \frac{\sqrt{2}}{2}, x = -1 - \frac{\sqrt{2}}{2}}$. Send me the next one 📸