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**Hint:**Here we will move the term with the variable on the right hand side of the equation. When you move any term from one side to opposite, the sign of the term also changes. Will use the factorization, and the square and square root concepts.

**Complete step-by-step solution:**

Take the given expression: $2{x^2} + 338 = 0$

Make the required term with the variable the subject and move the other term without the variable that is constant on the opposite side. When you move any term from one side to another, the sign of the term also changes. Positive terms become negative and vice-versa.

$2{x^2} = - 338$

Term multiplicative on one side if moved to the opposite side then it goes to the denominator.

${x^2} = - \dfrac{{338}}{2}$

Find the factors of the term on the numerator on the right hand side of the equation.

${x^2} = - \dfrac{{169 \times 2}}{2}$

Common factors from the numerator and the denominator cancel each other. Therefore remove from the numerator and the denominator.

$ \Rightarrow {x^2} = - 169$

Take square-root on both the sides of the above equation.

$ \Rightarrow \sqrt {{x^2}} = \sqrt { - 169} $

Square and square root cancel each other on the left hand side of the equation.

$ \Rightarrow x = \sqrt { - 169} $

The above equation can be re-written as:

$

\Rightarrow x = \sqrt {169 \times ( - 1)} \\

\Rightarrow x = \sqrt {169} \times \sqrt { - 1} \\

$

We know that square of negative term or the positive term always gives the positive term and $\sqrt {( - 1)} = i$

$ \Rightarrow x = \sqrt {{{( \pm 13)}^2}} \times i$

Square and square root cancel each other on the right hand side of the equation.

$ \Rightarrow x = \pm 13i$

This is the required solution.

**Note:**Square is the number multiplied itself and cube it the number multiplied thrice. Square is the product of same number twice such as ${n^2} = n \times n$ for Example square of $2$ is ${2^2} = 2 \times 2$ simplified form of squared number is ${2^2} = 2 \times 2 = 4$ . Know the difference between the squares and square roots and apply the concepts accordingly. Square is the number multiplied twice and square-root is denoted by $\sqrt {{n^2}} = \sqrt {n \times n} $ For Example: $\sqrt {{2^2}} = \sqrt 4 = 2$

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