Find All Complex Solutions 1/2x^4=648

$\frac{1}{2} x^{4} = 648$

Multiply each term in $\frac{1}{2} x^{4} = 648$ by $2$ to eliminate the fractions.

Multiply each term in $\frac{1}{2} x^{4} = 648$ by $2$ .

$\frac{1}{2} x^{4} \cdot 2 = 648 \cdot 2$

Simplify the left side.

Combine $\frac{1}{2}$ and $x^{4}$ .

$\frac{x^{4}}{2} \cdot 2 = 648 \cdot 2$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{x^{4}}{2} \cdot 2 = 648 \cdot 2$

Rewrite the expression.

$x^{4} = 648 \cdot 2$

Simplify the right side.

Multiply $648$ by $2$ .

$x^{4} = 1296$

Subtract $1296$ from both sides of the equation.

$x^{4} - 1296 = 0$

Factor the left side of the equation.

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

${(x^{2})}^{2} - 1296 = 0$

Rewrite $1296$ as $36^{2}$ .

${(x^{2})}^{2} - 36^{2} = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 36$ .

$(x^{2} + 36) (x^{2} - 36) = 0$

Simplify.

Rewrite $36$ as $6^{2}$ .

$(x^{2} + 36) (x^{2} - 6^{2}) = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 6$ .

$(x^{2} + 36) ((x + 6) (x - 6)) = 0$

Remove unnecessary parentheses.

$(x^{2} + 36) (x + 6) (x - 6) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} + 36 = 0$

$x + 6 = 0$

$x - 6 = 0$

Set $x^{2} + 36$ equal to $0$ and solve for $x$ .

Set $x^{2} + 36$ equal to $0$ .

$x^{2} + 36 = 0$

Solve $x^{2} + 36 = 0$ for $x$ .

Subtract $36$ from both sides of the equation.

$x^{2} = - 36$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 36}$

Simplify $\pm \sqrt{- 36}$ .

Rewrite $- 36$ as $- 1 (36)$ .

$x = \pm \sqrt{- 1 (36)}$

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{36}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{36}$

Rewrite $36$ as $6^{2}$ .

$x = \pm i \cdot \sqrt{6^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 6$

Move $6$ to the left of $i$ .

$x = \pm 6 i$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 6 i$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 6 i$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 6 i, - 6 i$

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

The final solution is all the values that make $(x^{2} + 36) (x + 6) (x - 6) = 0$ true.

$x = 6 i, - 6 i, - 6, 6$

Find All Complex Solutions 1/2x^4=648

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