Solve for x. 2x^(2)+1=0

Isolate x^2 term: Given the quadratic equation 2x^(2) + 1 = 0, we need to solve for x. First, we isolate the x^2 term by subtracting 1 from both sides of the equation. 2x^(2) + 1 - 1 = 0 - 1 2x^(2) = -1Divide by 2: Next, we divide both sides of the equation by 2 to solve for x^2. 2x^(2) / 2 = -1 / 2 x^2 = -1/2Take square root: To find x, we need to take the square root of both sides of the equation. Since we are taking the square root of a negative number, we will get complex solutions. x = ±√(-1/2)Use imaginary unit: We know that the square root of -1 is the imaginary unit i, so we can rewrite the solutions as: x = ±√(1/2) * i x = ±(1/√2) * iRationalize denominator: To rationalize the denominator, we multiply the numerator and the denominator by √2. x = ±(1/√2) * (√2/√2) * i x = ±(√2/2) * iFinal solutions: The final solutions to the equation 2x^(2) + 1 = 0 are: x = (±√2/2) * i

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Solve for $x$ . $\newline$ $2x^{2}+1=0$

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x

\newline

2x^{2}+1=0

Isolate $x^2$ term: Given the quadratic equation $2x^{2} + 1 = 0$ , we need to solve for x. $\newline$ First, we isolate the $x^2$ term by subtracting $1$ from both sides of the equation. $\newline$ $2x^{2} + 1 - 1 = 0 - 1$ $\newline$ $2x^{2} = -1$
Divide by $2$ : Next, we divide both sides of the equation by $2$ to solve for $x^2$ . $\newline$ $\frac{2x^{2}}{2} = \frac{-1}{2}$ $\newline$ $x^2 = -\frac{1}{2}$
Take square root: To find $x$ , we need to take the square root of both sides of the equation. Since we are taking the square root of a negative number, we will get complex solutions. $x = \pm\sqrt{\frac{-1}{2}}$
Use imaginary unit: We know that the square root of $-1$ is the imaginary unit $i$ , so we can rewrite the solutions as: $\newline$ $x = \pm\sqrt{\frac{1}{2}} \cdot i$ $\newline$ $x = \pm\left(\frac{1}{\sqrt{2}}\right) \cdot i$
Rationalize denominator: To rationalize the denominator, we multiply the numerator and the denominator by $\sqrt{2}$ . $x = \pm\left(\frac{1}{\sqrt{2}}\right) \cdot \left(\frac{\sqrt{2}}{\sqrt{2}}\right) \cdot i$ $x = \pm\left(\frac{\sqrt{2}}{2}\right) \cdot i$
Final solutions: The final solutions to the equation $2x^{2} + 1 = 0$ are: $\newline$ $x = (\pm\sqrt{2}/2) \cdot i$