Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Factor $u^{2} - 2 u - 3$ 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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 3.2

Write the factored form using these integers.

$(u - 3) (u + 1) = 0$

Step 4

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

$(x^{2} - 3) (x^{2} + 1) = 0$

Step 5

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

$x^{2} - 3 = 0$

$x^{2} + 1 = 0$

Step 6

Set $x^{2} - 3$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x^{2} - 3$ equal to $0$ .

$x^{2} - 3 = 0$

Step 6.2

Solve $x^{2} - 3 = 0$ for $x$ .

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Step 6.2.1

Add $3$ to both sides of the equation.

$x^{2} = 3$

Step 6.2.2

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

$x = \pm \sqrt{3}$

Step 6.2.3

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

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Step 6.2.3.1

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

$x = \sqrt{3}$

Step 6.2.3.2

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

$x = - \sqrt{3}$

Step 6.2.3.3

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

$x = \sqrt{3}, - \sqrt{3}$

Step 7

Set $x^{2} + 1$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Step 7.2

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

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Step 7.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 7.2.2

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

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

Step 7.2.3

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

$x = \pm i$

Step 7.2.4

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

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Step 7.2.4.1

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

$x = i$

Step 7.2.4.2

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

$x = - i$

Step 7.2.4.3

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

$x = i, - i$

Step 8

The final solution is all the values that make $(x^{2} - 3) (x^{2} + 1) = 0$ true.

$x = \sqrt{3}, - \sqrt{3}, i, - i$