Algebra Examples

$x^{4} + 3 x^{2} - 4 = 0$

Step 1

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

${(x^{2})}^{2} + 3 x^{2} - 4 = 0$

Step 2

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

$u^{2} + 3 u - 4 = 0$

Step 3

Factor $u^{2} + 3 u - 4$ using the AC method.

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Step 3.1

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 $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 3.2

Write the factored form using these integers.

$(u - 1) (u + 4) = 0$

Step 4

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

$(x^{2} - 1) (x^{2} + 4) = 0$

Step 5

Rewrite $1$ as $1^{2}$ .

$(x^{2} - 1^{2}) (x^{2} + 4) = 0$

Step 6

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 $b = 1$ .

$(x + 1) (x - 1) (x^{2} + 4) = 0$

Step 7

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

$x + 1 = 0$

$x - 1 = 0$

$x^{2} + 4 = 0$

Step 8

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 8.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 9

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 9.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 9.2

Add $1$ to both sides of the equation.

$x = 1$

Step 10

Set $x^{2} + 4$ equal to $0$ and solve for $x$ .

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Step 10.1

Set $x^{2} + 4$ equal to $0$ .

$x^{2} + 4 = 0$

Step 10.2

Solve $x^{2} + 4 = 0$ for $x$ .

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Step 10.2.1

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

Step 10.2.2

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

$x = \pm \sqrt{- 4}$

Step 10.2.3

Simplify $\pm \sqrt{- 4}$ .

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Step 10.2.3.1

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Step 10.2.3.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

Step 10.2.3.3

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

$x = \pm i \cdot \sqrt{4}$

Step 10.2.3.4

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

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

Step 10.2.3.5

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

$x = \pm i \cdot 2$

Step 10.2.3.6

Move $2$ to the left of $i$ .

$x = \pm 2 i$

Step 10.2.4

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

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Step 10.2.4.1

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

$x = 2 i$

Step 10.2.4.2

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

$x = - 2 i$

Step 10.2.4.3

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

$x = 2 i, - 2 i$

Step 11

The final solution is all the values that make $(x + 1) (x - 1) (x^{2} + 4) = 0$ true.

$x = - 1, 1, 2 i, - 2 i$