Algebra Examples

$x^{4} - x^{2} = x^{2} + 8$

Step 1

Move all the expressions to the left side of the equation.

Tap for more steps...

Step 1.1

Subtract $x^{2}$ from both sides of the equation.

$x^{4} - x^{2} - x^{2} = 8$

Step 1.2

Subtract $8$ from both sides of the equation.

$x^{4} - x^{2} - x^{2} - 8 = 0$

Step 2

Subtract $x^{2}$ from $- x^{2}$ .

$x^{4} - 2 x^{2} - 8 = 0$

Step 3

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

${(x^{2})}^{2} - 2 x^{2} - 8 = 0$

Step 4

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$u^{2} - 2 u - 8 = 0$

Step 5

Factor $u^{2} - 2 u - 8$ using the AC method.

Tap for more steps...

Step 5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 5.2

Write the factored form using these integers.

$(u - 4) (u + 2) = 0$

Step 6

Replace all occurrences of $u$ with $x^{2}$ .

$(x^{2} - 4) (x^{2} + 2) = 0$

Step 7

Rewrite $4$ as $2^{2}$ .

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

Step 8

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 $b = 2$ .

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

Step 9

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

$x + 2 = 0$

$x - 2 = 0$

$x^{2} + 2 = 0$

Step 10

Set $x + 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 10.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 10.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 11

Set $x - 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 11.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 11.2

Add $2$ to both sides of the equation.

$x = 2$

Step 12

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

Tap for more steps...

Step 12.1

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

$x^{2} + 2 = 0$

Step 12.2

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

Tap for more steps...

Step 12.2.1

Subtract $2$ from both sides of the equation.

$x^{2} = - 2$

Step 12.2.2

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

$x = \pm \sqrt{- 2}$

Step 12.2.3

Simplify $\pm \sqrt{- 2}$ .

Tap for more steps...

Step 12.2.3.1

Rewrite $- 2$ as $- 1 (2)$ .

$x = \pm \sqrt{- 1 (2)}$

Step 12.2.3.2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{2}$

Step 12.2.3.3

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

$x = \pm i \sqrt{2}$

Step 12.2.4

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

Tap for more steps...

Step 12.2.4.1

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

$x = i \sqrt{2}$

Step 12.2.4.2

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

$x = - i \sqrt{2}$

Step 12.2.4.3

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

$x = i \sqrt{2}, - i \sqrt{2}$

Step 13

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

$x = - 2, 2, i \sqrt{2}, - i \sqrt{2}$