Solve by Factoring 3x^4+15x^2-72=0

$3 x^{4} + 15 x^{2} - 72 = 0$

Step 1

Factor $3$ out of $3 x^{4} + 15 x^{2} - 72$ .

Factor $3$ out of $3 x^{4}$ .

$3 (x^{4}) + 15 x^{2} - 72 = 0$

Factor $3$ out of $15 x^{2}$ .

$3 (x^{4}) + 3 (5 x^{2}) - 72 = 0$

Factor $3$ out of $- 72$ .

$3 x^{4} + 3 (5 x^{2}) + 3 \cdot - 24 = 0$

Factor $3$ out of $3 x^{4} + 3 (5 x^{2})$ .

$3 (x^{4} + 5 x^{2}) + 3 \cdot - 24 = 0$

Factor $3$ out of $3 (x^{4} + 5 x^{2}) + 3 \cdot - 24$ .

$3 (x^{4} + 5 x^{2} - 24) = 0$

Step 2

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

$3 ({(x^{2})}^{2} + 5 x^{2} - 24) = 0$

Step 3

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

$3 (u^{2} + 5 u - 24) = 0$

Step 4

Factor $u^{2} + 5 u - 24$ 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 $- 24$ and whose sum is $5$ .

$- 3, 8$

Write the factored form using these integers.

$3 ((u - 3) (u + 8)) = 0$

Step 5

Factor.

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

$3 ((x^{2} - 3) (x^{2} + 8)) = 0$

Remove unnecessary parentheses.

$3 (x^{2} - 3) (x^{2} + 8) = 0$

Step 6

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

$x^{2} - 3 = 0$

$x^{2} + 8 = 0$

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 $x$ .

Add $3$ to 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.

$x = \pm \sqrt{3}$

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.

$x = \sqrt{3}$

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

$x = - \sqrt{3}$

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

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

Step 8

Set $x^{2} + 8$ equal to $0$ and solve for $x$ .

Set $x^{2} + 8$ equal to $0$ .

$x^{2} + 8 = 0$

Solve $x^{2} + 8 = 0$ for $x$ .

Subtract $8$ from both sides of the equation.

$x^{2} = - 8$

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

$x = \pm \sqrt{- 8}$

Simplify $\pm \sqrt{- 8}$ .

Rewrite $- 8$ as $- 1 (8)$ .

$x = \pm \sqrt{- 1 (8)}$

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{8}$

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

$x = \pm i \cdot \sqrt{8}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$x = \pm i \cdot \sqrt{4 (2)}$

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

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

Pull terms out from under the radical.

$x = \pm i \cdot (2 \sqrt{2})$

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

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

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.

$x = 2 i \sqrt{2}$

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

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

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

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

Step 9

The final solution is all the values that make $3 (x^{2} - 3) (x^{2} + 8) = 0$ true.

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

Solve by Factoring 3x^4+15x^2-72=0

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