Solve by Factoring x^4-4x^2-21=0

$x^{4} - 4 x^{2} - 21 = 0$

Step 1

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

${(x^{2})}^{2} - 4 x^{2} - 21 = 0$

Step 2

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

$u^{2} - 4 u - 21 = 0$

Step 3

Factor $u^{2} - 4 u - 21$ using the AC method.

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 $- 21$ and whose sum is $- 4$ .

$- 7, 3$

Write the factored form using these integers.

$(u - 7) (u + 3) = 0$

Step 4

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

$(x^{2} - 7) (x^{2} + 3) = 0$

Step 5

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

$x^{2} - 7 = 0$

$x^{2} + 3 = 0$

Step 6

Set $x^{2} - 7$ equal to $0$ and solve for $x$ .

Set $x^{2} - 7$ equal to $0$ .

$x^{2} - 7 = 0$

Solve $x^{2} - 7 = 0$ for $x$ .

Add $7$ to both sides of the equation.

$x^{2} = 7$

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

$x = \pm \sqrt{7}$

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.

$x = \sqrt{7}$

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

$x = - \sqrt{7}$

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

$x = \sqrt{7}, - \sqrt{7}$

Step 7

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

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

$x^{2} + 3 = 0$

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

Subtract $3$ from both sides of the equation.

$x^{2} = - 3$

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

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

Simplify $\pm \sqrt{- 3}$ .

Rewrite $- 3$ as $- 1 (3)$ .

$x = \pm \sqrt{- 1 (3)}$

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{3}$

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

$x = \pm i \sqrt{3}$

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.

$x = i \sqrt{3}$

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

$x = - i \sqrt{3}$

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

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

Step 8

The final solution is all the values that make $(x^{2} - 7) (x^{2} + 3) = 0$ true.

$x = \sqrt{7}, - \sqrt{7}, i \sqrt{3}, - i \sqrt{3}$

Solve by Factoring x^4-4x^2-21=0

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