Solve by Factoring x^4-5x^2=3x^2+48

$x^{4} - 5 x^{2} = 3 x^{2} + 48$

Step 1

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

Subtract $3 x^{2}$ from both sides of the equation.

$x^{4} - 5 x^{2} - 3 x^{2} = 48$

Subtract $48$ from both sides of the equation.

$x^{4} - 5 x^{2} - 3 x^{2} - 48 = 0$

Step 2

Subtract $3 x^{2}$ from $- 5 x^{2}$ .

$x^{4} - 8 x^{2} - 48 = 0$

Step 3

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

${(x^{2})}^{2} - 8 x^{2} - 48 = 0$

Step 4

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

$u^{2} - 8 u - 48 = 0$

Step 5

Factor $u^{2} - 8 u - 48$ 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 $- 48$ and whose sum is $- 8$ .

$- 12, 4$

Write the factored form using these integers.

$(u - 12) (u + 4) = 0$

Step 6

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

$(x^{2} - 12) (x^{2} + 4) = 0$

Step 7

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

$x^{2} - 12 = 0$

$x^{2} + 4 = 0$

Step 8

Set $x^{2} - 12$ equal to $0$ and solve for $x$ .

Set $x^{2} - 12$ equal to $0$ .

$x^{2} - 12 = 0$

Solve $x^{2} - 12 = 0$ for $x$ .

Add $12$ to both sides of the equation.

$x^{2} = 12$

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

$x = \pm \sqrt{12}$

Simplify $\pm \sqrt{12}$ .

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$x = \pm \sqrt{4 (3)}$

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

$x = \pm \sqrt{2^{2} \cdot 3}$

Pull terms out from under the radical.

$x = \pm 2 \sqrt{3}$

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.

$x = 2 \sqrt{3}$

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

$x = - 2 \sqrt{3}$

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

$x = 2 \sqrt{3}, - 2 \sqrt{3}$

Step 9

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

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

$x^{2} + 4 = 0$

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

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

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

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

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

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

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

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

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

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

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

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

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

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

$x = \pm i \cdot 2$

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

$x = \pm 2 i$

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.

$x = 2 i$

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

$x = - 2 i$

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

$x = 2 i, - 2 i$

Step 10

The final solution is all the values that make $(x^{2} - 12) (x^{2} + 4) = 0$ true.

$x = 2 \sqrt{3}, - 2 \sqrt{3}, 2 i, - 2 i$

Solve by Factoring x^4-5x^2=3x^2+48

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